Is Mathematics Invented or Discovered?

March 7, 2017 1:00 PM

Some people think that to make an advance in mathematics is to invent something (perhaps it is to invent new uses for language). Others think that to make an advance in mathematics is to discover something (perhaps it is to discover facts about abstract objects that live in Plato's heaven). In Is Mathematics Invented or Discovered?, Season 7, Episode 9 of the interview documentary series Closer to Truth, Robert Lawrence Kuhn talks with Roger Penrose, Mark Balaguer, Gregory Chaitin, Stephen Wolfram, and Frank Wilczek about whether mathematics is invented or discovered.

Some of the interviews, as well as interviews on the same or a similar theme, are available in stand alone versions:

Roger Penrose

Gregory Chaitin

Stephen Wolfram

Paul Davies

David Gross

George Lakoff

Max Tegmark

Steven Weinberg

One common point of reference for the discussion -- and what is probably required reading for anyone seriously interested in the topic -- is Eugene Wigner's paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences (pdf).

For those of you who want more, there is also a NOVA episode on The Great Math Mystery.

And if you want more interviews on other interesting philosophical, scientific, and religious topics, there is more from Closer to Truth.

posted by Jonathan Livengood (133 comments total)

It's the discovery of an invention.
posted by sammyo at 1:05 PM on March 7, 2017 [1 favorite]

I am here to protest against the automatic identification of any purported realm of abstract objects with Plato's cryptic theory of the εἶδος.
posted by thelonius at 1:06 PM on March 7, 2017 [13 favorites]

...or y'know, maybe both? A big chunk of mathematics is invented tools and ways of analyzing, but part of what they are manipulating maps to fundamental things about the universe.
posted by tavella at 1:08 PM on March 7, 2017 [11 favorites]

My take on the effectiveness of mathematics is the farmer and the barn - the side of his barn is covered with targets, with bullet holes in the bullseye of every target. The farmer is a poor marksman and he paints a target around each bullet hole.

Our concrete knowledge of things tends to follow the availability of mathematical tools to express them.
posted by idiopath at 1:11 PM on March 7, 2017 [16 favorites]

The idea of inventing concepts is anthropocentric. All we are doing is slowly discovering new permutations of what is possible, and everything that isn't a fundamental concept is derived from fundamental concepts. That is my entirely uninformed opinion.
posted by grumpybear69 at 1:11 PM on March 7, 2017 [3 favorites]

Dunno about mathematics, but algorithms are definitely invented.
posted by infinitewindow at 1:14 PM on March 7, 2017 [2 favorites]

tavella: if that were true, the majority of mathematical research would be applicable to real world problems. The truth is the opposite - if you find a problem there's probably a way to map it to something mathematical, but the majority of mathematical research doesn't apply to any problem anyone has found yet (outside math itself, of course).
posted by idiopath at 1:14 PM on March 7, 2017 [1 favorite]

sammyo: "It's the discovery of an invention."

Or the invention of a language to describe a discovery?
posted by chavenet at 1:15 PM on March 7, 2017 [10 favorites]

I have never thought of it that way.


And after spending some time thinking about it, I'll stop, lest I have to pick bits of my mind from all over the room.
posted by lmfsilva at 1:17 PM on March 7, 2017 [3 favorites]

When working on/thinking about mathematics, sometimes it strongly feels like you're discovering something, sometimes it strongly feels like you're inventing something, and the former is much more gratifying than the latter. This is not mathematical, or metamathematical, though: it's just a description of how that shit feels on your skin.

Right now, and much of the time, mathematics isn't something I'm discovering or inventing. It's just a source of frustration/angst. Other things would be less frustrating but even more angst-ridden, so one comes back to maths a few times a day.
posted by busted_crayons at 1:26 PM on March 7, 2017 [11 favorites]

This topic is mentioned in a 3blue1brown video about infinite sums.
posted by a snickering nuthatch at 1:29 PM on March 7, 2017 [1 favorite]

I am here to protest against the automatic identification of any purported realm of abstract objects with Plato's cryptic theory of the εἶδος.

I did preface with a "perhaps" -- intending to give the Platonic theory as one way of giving sense to the claim that mathematics is discovered. :)
posted by Jonathan Livengood at 1:37 PM on March 7, 2017 [4 favorites]

the majority of mathematical research doesn't apply to any problem anyone has found yet (outside math itself, of course).

Is this research being done in preparation for a day when it might be useful, or is it just math for math's sake? (Just curious, not trying to slag on math or anything!)
posted by showbiz_liz at 1:40 PM on March 7, 2017

showbiz_liz: research in 'pure math' is very much math for math's sake. 'applied math' is a different basket of apples.
posted by crazy_yeti at 1:42 PM on March 7, 2017

Or the invention of a language to describe a discovery?

I really like this formulation. It seems most likely to be adjacent to the truth.
posted by tobascodagama at 1:46 PM on March 7, 2017 [4 favorites]

It seems kind of weird that in what's basically a presentation on Philosophy of Mathematics 101 they talk to almost no philosophers of mathematics.
posted by Sangermaine at 1:48 PM on March 7, 2017 [19 favorites]

I am here to protest against the automatic identification of any purported realm of abstract objects with Plato's cryptic theory of the εἶδος.

Doing Logos' work.

I have personally often wrestled with the question of whether it makes any sense to say that, for example, irrational numbers are real things (haha, "real"). My instinct is that, yes, they must be, because circles and right triangles are things we can experience in the world, and those necessitate pi and root-2. So, discovery, right? But then, what if circles and triangles are themselves human inventions?
posted by 256 at 1:54 PM on March 7, 2017 [4 favorites]

That's an interesting, but lopsided selection of interviewees. I don't see any bio-/geo-'atmospheric physicists or others who would (1) express their fascination at the increasing level of specificity that mathematics can reach in biological/ecological/climatological systems (2) while horrifying theoretical and particle physicists at how comfortable they are with the influence of chaos on these systems and the possibility that mathematics might not be the discovery or invention that can anticipate these things very deeply.

Thanks for sharing. I could listen to people talk about these questions all day (especially when the discussion is set in what might be a Victorian grandmother's library, filmed through a Vaseline-coated soft focus lens, like the Lakoff one. Such romance!)
posted by late afternoon dreaming hotel at 1:56 PM on March 7, 2017 [2 favorites]

Is this research being done in preparation for a day when it might be useful, or is it just math for math's sake? (Just curious, not trying to slag on math or anything!)

IME, pure math works a lot like a giant conversation. It's math for math's sake, but the constraints and motivations feel real largely because the target audience is your friends. Like: it's an art form where the pieces of art being produced are often themselves tools that your friends can use to make their art, etc.

Sometimes you do something because it's there, or enticing, even if nobody else is unlikely to ever use it, but usually it's good to limit that sort of behaviour to cases where: (a) it will put to rest something that lots of your friends have also been bothered by (even if it's not the sort of thing they can build on), or (b) to situations where you really can't help yourself. Mostly one (or I, anyway) is most gratified by doing something that someone else can engage with and twist to their own nefarious ends.

If I weren't part of the math community, I'd probably still think about math (increasingly less productively, as I got more and more out of touch), but I wouldn't bother doing math in the same kind of organised way, defining things carefully, proving theorems, etc. I'd probably just draw blobs.

On the other hand, pure math ends up being connected to other fields (i.e. "useful") in unpredictable ways, surprisingly frequently, and it's very interesting to think about why.
posted by busted_crayons at 1:59 PM on March 7, 2017 [11 favorites]

Excellent post. It will take a while to get through all of the links, but I remember many times in grad school discussing this with my spouse (who also studied math) and I always fell on the side of discovery.

But then, what if circles and triangles are themselves human inventions?

Imagine our universe immediately before the first human. There are many objects or relations that still exist with properties of circles and triangles, so yeah, I'd say they are not human inventions.
posted by noneuclidean at 2:00 PM on March 7, 2017 [2 favorites]

It seems like a lot of the struggle is not in finding new information but in organizing it in a form that will fit into the human skull.
posted by Flexagon at 2:06 PM on March 7, 2017 [5 favorites]

Maybe mathematics was in our hearts all along?
posted by Kabanos at 2:12 PM on March 7, 2017 [12 favorites]

The true mathematics was the friends we made along the way.
posted by tobascodagama at 2:26 PM on March 7, 2017 [22 favorites]

Or the invention of a language to describe a discovery?

There is definitely an important distinction to be made between mathematics and the words and symbols we use to talk about mathematics. I definitely consider those words and symbols to be invented, but whether the underlying mathematics is discovered or invented is the much harder question.
posted by noneuclidean at 2:27 PM on March 7, 2017 [3 favorites]

Who discovered* who?



*this is a joke** based on the "Who rescued who?" bumper stickers about animal adoption, which should actually read, "Who rescued whom?" N.B. there are some grammatical problems with the joke due to the fact that mathematics are probably not sentient in a way that would make them appropriate for the "who" pronoun.

**a very funny one

That Wigner paper was a fun read.

I'll show myself out.
posted by radicalawyer at 2:32 PM on March 7, 2017 [2 favorites]

I think mathematics is just a translation tool between actual reality and our limited ability to understand reality.
posted by rocket88 at 3:05 PM on March 7, 2017 [3 favorites]

Most recently I've been influenced by Richard Hamming's thoughts on this question.
posted by grobstein at 3:20 PM on March 7, 2017 [5 favorites]

I am here to protest against the automatic identification of any purported realm of abstract objects with Plato's cryptic theory of the εἶδος.

Gesundheit.
posted by Splunge at 3:29 PM on March 7, 2017 [1 favorite]

I have personally often wrestled with the question of whether it makes any sense to say that, for example, irrational numbers are real things (haha, "real"). My instinct is that, yes, they must be, because circles and right triangles are things we can experience in the world, and those necessitate pi and root-2. So, discovery, right? But then, what if circles and triangles are themselves human inventions?

Before humans existed the cross-sections of stars and planets were fucked-up-a-gons rather than figures with a circumference-to-diameter ratio that approaches π.
posted by XMLicious at 3:37 PM on March 7, 2017 [2 favorites]

So, what's up with mathematical intuition?
posted by The Gaffer at 3:41 PM on March 7, 2017

But seriously, something else to observe in the conversation (which probably appears in one of the OP links) is that the relationship which π represents isn't just a property of circles and right triangles, but is fundamental (in human terms) to trigonometry, and thereby also fundamental to wave mechanics, and to the behavior of vibrations in matter, electromagnetic waves, the quantum wave function that describes all of that non-general-relativistic phenomena, the music of the spheres and the Song of Ilúvatar...

So if it's just an optical illusion, as it were, which is an artifact of human perception of the world, that is one hell of an optical illusion.
posted by XMLicious at 3:51 PM on March 7, 2017 [2 favorites]

With the disclaimer that I only know enough about this sort of philosophy to be dangerous and could be abusing basically every term I invoke after this clause, I always feel like "invention" is under-specified in this discussion. What does it mean, and what are the pre-conditions required for it to mean that? Not that I endorse this view or make any assertions about its internal consistency, but you could imagine a worldview that holds that all concepts are arrangements of a (potentially countably infinite) set of symbols*, and so are themselves enumerable, meaning that any conceptual invention is a just a selection of some natural number put through a pairing function. That's not to trivialize the process of such a selection, just to point out that, depending on what you already hold to be true about the universe, "invention" seems like it can diverge from its intuitive sense, to the extent that it basically sounds like discovery.

* I guess if we admit that different sets of symbols can be said to represent the same concept, then "concepts" are maybe better represented as sets of those sets whose members satisfy some predicate, but I have the vague sense that this definition probably allows for some unintended problems in the vein of Russell's paradox, but it's besides the point anyway.
posted by invitapriore at 4:37 PM on March 7, 2017 [4 favorites]

I guess if we admit that different sets of symbols can be said to represent the same concept, then "concepts" are maybe better represented as sets of those sets whose members satisfy some predicate, but I have the vague sense that this definition probably allows for some unintended problems in the vein of Russell's paradox, but it's besides the point anyway.


I think the early Russell did indeed believe in the actual reality of propositions, and had some sort of view where sentences can be said to have the same meaning because they express the same proposition, but I think he moved away from that. It's an interesting question if things like set theory paradoxes had anything to do with that; I believe that the status of sentences like "The present King of France is bald", which he invented his theory of descriptions to eliminate, were the main problem for him.
posted by thelonius at 4:49 PM on March 7, 2017 [2 favorites]

Also, I've only gotten through the Penrose interview so far, so maybe someone answers this, but: what does the "invent" side hold to be the source of the constraints on what statements follow from the invention's formulation? I can't think of a sense in which we invent those, and then what "invention" means in the face of the fact that something invented is subject to laws that existed prior to it and weren't formulated in terms of its existence seems murky to me.
posted by invitapriore at 4:50 PM on March 7, 2017 [1 favorite]

That objects exist is a feature of the universe; that we group them into sets is a creation of our intellect. That some of these sets are similar (i.e., each set's elements can be put in a 1:1 relationship with the other set's elements) is a feature of the universe; that we identify this similarity as a number is a creation of our intellect. The same goes for every other relationship that we recognise: the objects and circumstances and events that give rise to them are inherent to the universe, but the meaning we impose upon them is a creation of our minds.

We can use addition to add the quantities of two groups of atoms, or groups of countries, or groups of hours; but that doesn't mean that there is an intrinsic two-ness about them, or that they are inherently addable. We are the ones who recognise the elements of the sets, the sets themselves, and the superset that combines them. The same is true of all mathematical relationships that we discern in real things: we are the ones who recognise and give significance to these relationships, and mathematics is the language which we use to describe it.
posted by Joe in Australia at 5:18 PM on March 7, 2017 [2 favorites]

but that doesn't mean that there is an intrinsic two-ness about them, or that they are inherently addable.

So why does it work consistently even in cases where the application wasn't understood to start with? There must be some common feature that unites all triangles, even if it only exists as a metaphysical property we can't easily identify among the physical stuff, but there's some kind of consistency to the patterns found among these different phenomena that can be analyzed in terms of, say, "two-ness" or else that conceptual technology wouldn't be useful.

Metaphysics is the domain of philosophy that deals with the meta rules that govern observable physical phenomena, isn't it? We know there are forces and other real physical phenomena that aren't reducible to simple atomic constituents and that can only be observed indirectly through their effects.

I think it's likely things like "two-ness" and "triangle-ness" are real, meaningful properties of physical objects at the metaphysical level of analysis--I.e., they concern the patterns and arrangements of ideas relative to each other, not merely the more naively observable physical features of objects viewed as atomic units. It's like the color red: at a certain level, in a hypothetical universe with no conscious beings, observable color might be irrelevant and not even a meaningful property of an object. It doesn't much matter if a t-shirt is red or blue, to a colorblind consciousness. And redness or blueness as specific physical realities are created in the mind, but the patterns of relationships among objects that color allows us to describe can be meaningful and useful, even though they're not as physically "real" or obvious as they might seem to a naive realist.
posted by saulgoodman at 5:34 PM on March 7, 2017 [1 favorite]

The best thing about π is that the integral over the real line of exp(-x^2) = sqrt(π).

I understand the proof, and can repeat the steps, but it still amazes me that it should be true. Given that the π arises out of a trick to evaluate the integral, it seems like a mathematical coincidence of the type Chaitin talks about.
posted by claudius at 5:37 PM on March 7, 2017 [2 favorites]

... what does the "invent" side hold to be the source of the constraints on what statements follow from the invention's formulation? I can't think of a sense in which we invent those, and then what "invention" means in the face of the fact that something invented is subject to laws that existed prior to it and weren't formulated in terms of its existence seems murky to me.

One way an "invent" person might go is to say that the constraint comes from a decision to follow certain rules. That is to say that it's all invention: invention all the way down. We invent axioms and definitions. We also invent the rules of inference. We could have invented other rules of inference (and in fact, we have done so).

You might wonder: What gives the rules their force? Well, what gives the rules of chess their force (i.e. why can't I move my queen like a knight or promote my pawns immediately or ...)? The answer is the same for mathematics as it is for any case of rule-following. (Or at least, this is how I imagine at least some proponents of the "invent" position going. More specifically, this is how I imagine Wittgenstein going.)
posted by Jonathan Livengood at 5:40 PM on March 7, 2017 [3 favorites]

We'll meet aliens and we'll be like, "Mathematics!" and they'll be like, "what the hell are you talking about"
posted by XMLicious at 5:42 PM on March 7, 2017 [1 favorite]

I don't know about mathematics, but algorithms are definitely discovered.
posted by erniepan at 6:16 PM on March 7, 2017 [1 favorite]

One way an "invent" person might go is to say that the constraint comes from a decision to follow certain rules. That is to say that it's all invention: invention all the way down. We invent axioms and definitions. We also invent the rules of inference. We could have invented other rules of inference (and in fact, we have done so).

That's a good point. That kind of crossed my mind, but I dismissed it with the thought that systems of inference are similar in that they seem to be bound by something other than just the entities that they specify, but I guess that's not true for e.g. predicate logic, which is only bound by rules that it states itself. The situation seems hairier with something like first-order logic, which seems to depend on the properties of sets for its force? But maybe I just don't know enough about the foundations of that relationship, and it seems like a pretty convincing argument otherwise. I guess the best angle of attack for a believer in discovery from there, then, is to appeal to "consonances" between the physical and the mathematical like Wigner does?
posted by invitapriore at 6:18 PM on March 7, 2017

I would say that mathematics are invented, being a product of human image and pattern recognition proclivities, but the useful relationship between mathematics and reality is discovered. But I am not a philosopher of math, so whatevs.
posted by Bringer Tom at 6:26 PM on March 7, 2017 [1 favorite]

Oh man, though Hamming really deviously undermines the apparent meaning of those consonances in that essay that grobstein links. I'm amazed that anyone ever comes away from this question with a stable opinion on the matter.
posted by invitapriore at 6:31 PM on March 7, 2017 [1 favorite]

256,

Circles, triangles, and sqrt(2) don't exist in nature. You will never find a perfect circle in nature, not one that holds up as a circle below the quantum/Planck scale. What is happening is what XMLicious describes. We approximate shapes that are actually very complicated with circles and triangles because those shapes are easier to understand. Irrational numbers are basically rounding error in reverse.

previously.
posted by yeolcoatl at 6:36 PM on March 7, 2017 [2 favorites]

Banach Tarski is another good example. If you think of the 3-ball as a real physical object, the result makes no sense. If you think of the 3-ball as an object of infinite density, the result makes perfect sense. It's basically just Hilbert's hotel in 3d.
posted by yeolcoatl at 6:47 PM on March 7, 2017 [2 favorites]

Previously. (NB: Only if your mind isn't already blown.)
posted by ZenMasterThis at 6:47 PM on March 7, 2017

Forget defining "invention", what do we mean by "mathematics"? Are we talking about the numbers, equations, symbols, and terms used to describe relationships? Or are we talking about the relationships themselves? If the former, it's trivially obvious that it's an invention -- all language is an invention. If you discover the moon and decide to call it "Moon", you just invented the word "Moon" -- no one would say you are "discovering" the English language, even if the moon itself was a discovery.

So we must be talking about the latter, the relationships themselves. Some of those exist in nature whether we describe them with a language or not. (If we have two sheep and add one there will be three sheep no matter whether or not we have terms for "two", "three", "one", or "add"). I think many of those could reasonably said to be discovered, with a language to describe it invented after the fact. On the other hand, once we have that language, we can play with it and make relationships that do not exist in nature, so we might reasonably call those invented.

Hm, I guess you could argue that all abstract concepts exist in some a priori way and are simply waiting to be discovered, but that line of thinking means that invention becomes a meaningless concept. We go back to "discovering" the English language when you make up a word. That seems silly, it's redefining the terms to mean what you want them to mean.

So I'm going to go with -- both. Some mathematics is discovered, and some is invented.
posted by kyrademon at 6:54 PM on March 7, 2017 [4 favorites]

The question sounds like a free will vs determinism debate in a different guise. Maybe the situation is that if you could answer one question you could answer the other.

Second, it sounds old school to have the discourse revolve around invention, because some other people attempt to subscribe to the idea that math (or science) is a social construct (see: white people debating/handwringing over Mochizuki). That notion of "This is socially constructed therefore practitioner should should behave in certain ways" is a can of worms, as I'd claim it's a terrific example of science people misusing/oversimplifying ideas from critical theory and related areas/communities of continental philosophy.

I've seen a snippet of this series before and was left confused by the interviewer, and actually thought he was some sort of crackpot, but now that I know the context better it sounds like an interesting documentary project.
posted by polymodus at 8:13 PM on March 7, 2017 [2 favorites]

You will never find a perfect circle in nature, not one that holds up as a circle below the quantum/Planck scale. What is happening is what XMLicious describes.

I was actually trying to convey agreement with 256: panta rhei; in the same way rivers exist only as patterns, and to the extent that anything can be said to exist at all it's only as a pattern, circles also exist as patterns, as average convergences of the variations over time in real phenomena like bodies of matter formed by inverse-square-law forces and radiating waves of light. Similarly I'm sure that by the same reasoning one could say that no oscillation forms a perfect sine wave or composition of trigonometric functions, but that does not seem adequate cause to me to say that sine waves exist only in a less real way than the light or orbits or vibrations.

And how do I know that my left shoe is an object, rather than merely a hole in the not-my-left-shoe? I might have to disagree with "That objects exist is a feature of the universe".

By the way none of us are individuals, but merely fourth-dimensional tentacles of the post-singularity super-organism reaching back in time. Iä, iä, Cthulhu fhtagn.

I was thinking, the oldest thing I've heard referred to as a "hammer" is a rock with a particular shape which a pre-Homo-sapiens-sapiens human used to hit things. So the invention of the hammer occurred when someone picked up a rock and hit something with it.

I guess I could go for saying mathematics is that kind of invention.
posted by XMLicious at 8:22 PM on March 7, 2017 [1 favorite]

Forced-choice dichotomies: Bad questions, or worst questions?

(My 2¢: Mathematics is the study of discovering the consequences of choosing to follow some set of rules, as well as the invention of techniques for facilitating both the act of discovery and the understanding of the consequences).
posted by belarius at 8:46 PM on March 7, 2017 [2 favorites]

that doesn't mean that there is an intrinsic two-ness about them, or that they are inherently addable.

So why does it work consistently even in cases where the application wasn't understood to start with? There must be some common feature that unites all triangles [...]

It's a trick, like those performers that ask you to add numbers and reverse them and so forth, and then tell you the number you first thought of.

When we recognise things as being countable, it means that we can describe them with the language we use for other countable objects. If we recognise something as being triangular, it means that we can describe it with the language we use for geometric shapes - we can talk about its area or perimeter or what have you, even if it isn't a real actual triangle and is just a set of three coordinates. The same goes for other things that can be modeled mathematically: when we say that bacteria have a geometric growth rate, we don't mean that they possess something called a "geometric growth rate"; we simply mean that since each bacterium can reproduce itself after so many hours, the number of bacterium can be estimated by an exponential formula. We would know that even if we had never seen bacteria, just from being asked about entities that multiply themselves on a fixed schedule.

So if someone says it's remarkable that trigonometric identities show up in (e.g.) the passage of light, it isn't because photons carry little slide rules or computers that help them decide what to do. It's because the behaviour we're observing depends on the relationship between light and the fields that it moves through, and we can talk about that relationship using trigonometric language.
posted by Joe in Australia at 9:22 PM on March 7, 2017

Second, it sounds old school to have the discourse revolve around invention, because some other people attempt to subscribe to the idea that math (or science) is a social construct (see: white people debating/handwringing over Mochizuki). That notion of "This is socially constructed therefore practitioner should should behave in certain ways" is a can of worms, as I'd claim it's a terrific example of science people misusing/oversimplifying ideas from critical theory and related areas/communities of continental philosophy.

Can you explain a little more what you mean by this? I know Mochizuki is a mathematician who has made some significant claims and done work that might be very innovative, except that his body of work is so self-contained and divergent from well-known math that pretty much nobody else understands it well enough to confirm his claims. But I'm not sure how this situation relates to the rest of what you're saying here.
posted by atoxyl at 10:12 PM on March 7, 2017 [1 favorite]

It's a trick, like those performers that ask you to add numbers and reverse them and so forth, and then tell you the number you first thought of.

When we recognise things as being countable, it means that we can describe them with the language we use for other countable objects[...]

So if someone says it's remarkable that trigonometric identities show up in (e.g.) the passage of light, it isn't because photons carry little slide rules or computers that help them decide what to do. It's because the behaviour we're observing depends on the relationship between light and the fields that it moves through, and we can talk about that relationship using trigonometric language.

But some phenomena can be related to trigonometry and its various isomorphisms, while other phenomena can be related to countable sets, or a different class of mathematical structures. What makes an object's existence or not a real property, or for example its location relative to other objects in its inertial rest frame a real property if you accept that as such, but the property of whether it can be described with a particular type of mathematics a trick rather than something which is real?

Are countable objects uncountable when we're not looking? Or do they relate to other parts of mathematics when we're not describing them?

Does the ratio between the circumference of a cross-section of Jupiter and its diameter converge on a number other than π when it's not being described, or when no one exists who can describe it?

Presumably objects continue to exist when we're not describing them as existent. Whatever kind of slide rule is necessary to measure whether or not they exist, they appear to be carrying that one.

Maybe it's the universe itself which is carrying the whether-it-exists slide rule, but is bad at math and doesn't carry the mathematical properties slide rule.
posted by XMLicious at 10:19 PM on March 7, 2017

What makes an object's existence or not a real property, or for example its location relative to other objects in its inertial rest frame a real property if you accept that as such, but the property of whether it can be described with a particular type of mathematics a trick rather than something which is real?

Because counting is something that happens in our minds. Planets are countable because we observe and count them, but it's a property that we impose on the planets and not one that they possess in and of themselves. Planets (used to) have mystical significance too, but I hope you wouldn't say that Venus is intrinsically female or that Jupiter has anything to with royalty. When I say that "countability" is a trick, incidentally, I don't mean that we're being at all false about it, it's just that our idea of "countability" is essentially self-referential: we define a set of things and that makes it countable. There haven't always been nine (seven! ten! eight! ten!) planets; we changed our definitions but that didn't change the astronomical bodies.
posted by Joe in Australia at 1:29 AM on March 8, 2017 [1 favorite]

My point is less about the physical reality of "triangle-ness" than to get at the idea syntax and pattern arrangements are fundamental to the human sense of meaning and imperfect ability to negotiate reality. I think the observed evidence that math does yield useful generalizable results is sufficient for me to be convinced its utility isn't purely accidental. Math just offers a set of incredibly precise and more rigorously thought out conventions for organizing and simplifying complex ideas about patterns of relationships. Relationships seem like a human construct, but physical reality does seem to care about organization and order in the sense that the same basic raw physical materials might or might not behave certain observable ways depending on their arrangements. The logic and structure of human thought processes at some level must be constrained by the same physical laws as anything else in the universe, if there are universal natural laws, so human thinking doesn't happen in a vacuum and isn't a process that occurs in isolation from physical reality. No matter how highly we think of imagination and creativity, there must be ideas we're physically incapable of understanding; if mathematical language and the refinement of mathematical intuition as a collaborative enterprise produce useful results that seem meaningful and work in practice to describe reality that have predictive power, something we're doing is less invention and more discovery, in terms of that framing. Those conceptual categories are probably too narrow and themselves both "invented" but allowing for the imperfect fit, I'd say it's not impossible even pure reasoning could yield descriptions of real metaphysical phenomena--implicit rules of syntax, larger patterns of organization, etc. Basically all the stuff of the universe that doesn't fit neatly into the universe as a composite of objects metaphor (which as XMLicious points out correctly is not fundamental--the universe isn't made of little round balls, it's made of patterns of relationships and forces that interact to create effects that we interpret in simpler, semi-fictional terms (like the object model).

Tl;dr: I'm still in the camp that thinks the generalizeable stuff math accidentally discovers might be real in ways that are less intuitive when you're working in less rigorous and precise languages.

The interesting thing is that language as a technology itself can drive discovery and useful insight and further understanding but it turns out the degree of precision and rigor in a particular language for some reason seems to matter a lot.
posted by saulgoodman at 4:08 AM on March 8, 2017 [2 favorites]

It's all about pattern. As far as we know, the observable universe behaves at heart as a set of things that interact in repeatable, immutable ways. It's a circular definition, because we define things by their rules and rules by their things, but nonetheless there is a something that does something, and both somethings seem to be and do what it/they is/are in repeatable, immutable fashion. And there are discrete patterns of behaviour, so there is differentiation.

Again as far as we can tell, this has been the case for some time, certainly enough for the things to do their thing repeatedly. A differented universe that repeats has some nature of division and repetition, and to me this is the mathematical nature of the universe.

After quite a lot of this business, one consequence was temporary patterns of differentiation and repetition that could hold local traces of past events - to some extent, the whole universe is thought to do that all the time, but these temporary patterns could - had to - hold specific traces, which acted on those temporary patterns to encourage those patterns (and the traces they held) own repetitions. In other words, life started and started to evolve, and it is perhaps best characterised by its ability (or necessity) to abstract patterns from around it which furthers its own continuation. (I'm comfortable with not making a difference between patterns, material and energy here, although that's another story.)

After yet more time (or repetition in space, or however you want to call it), some life's ability to abstract patterns reached the stage we're at now, where we can manipulate our own complex models in ways that are not directly caused by outside patterns, but which can occupy places in our models as if they did. Other life can do this, to various extents, but we are apparently much more skilled at self-manipulation in our internal pattern-trace-store. And we are the best mathematicians. No coincidence (and no disrespect to crows).

Pattern usage infers matching and recognition, and we find patterns useful once they are recognised. Whether the matrix for that recognition comes from outside ourselves (discovery) or 'inside' ourselves (invention) is always going to be a fuzzy and syncretic business, and there cannot be a clear dividing line. Not all inventions match external patterns in ways that reflect an actual external pattern, and I feel that most of the experience of being human (as opposed to being elsewhere on the scale of observed sentience) is a consequence of that. Certainly philosophy, science and Metafilter.

So the invention/discovery question in mathematics is profoundly linked to our innate nature, and cannot be determined without further evolution of our self-awareness. Whether it will contribute to that evolution - and whether it's profoundly important or trivially notable - I have not scooby number one.
posted by Devonian at 4:30 AM on March 8, 2017

I might have to disagree with "That objects exist is a feature of the universe".

That would make you one quarter Buddhist
posted by doiheartwentyone at 4:32 AM on March 8, 2017

That would make you one quarter Buddhist

Only if you elide compound objects and phenomena with all objects. But here we stray into the Tao of Physics, and while that has a history of creating grafitfyingly exothermic reactions among the particle physicists, it has yet to evince other useful developments.
posted by Devonian at 4:50 AM on March 8, 2017 [1 favorite]

And how do I know that my left shoe is an object, rather than merely a hole in the not-my-left-shoe? I might have to disagree with "That objects exist is a feature of the universe".

That something exists is observably a feature of the universe, even if the observer chooses to avoid making the fundamental distinction between that-which-observes and that-which-is-observed.

As soon as we begin to say anything about that which exists, we inevitably break it into conceptually separable parts and make such distinctions as are convenient for the labelling of those parts and the description of their relationships.

Your left shoe is both an object and a hole in not-your-left-shoe, but it's the distinction you choose to draw between your left shoe and not-your-left-shoe that makes that true. And that distinction is purely conceptual. Observe the region you've decided marks the boundary between your left shoe and not-your-left-shoe closely enough, and you will always find a scale at which there is no clearly correct path along which to slice the conceptual knife that divides the one from the other.

Isolation of physical objects is only ever approximate. Much of the time it's an excellent approximation, but it's never as completely clean as the various kinds of boundary inherent in mathematical models of those objects.
posted by flabdablet at 6:04 AM on March 8, 2017

fladablet: Exactly right I think. And that understanding squares with Incompleteness. Larger patterns and organization of function are much more important and powerful levels of analysis than more reductive analysis, but they're also much fuzzier and harder to conceptualize and intuit. But intuition, which is still very poorly understood, turns out to be at least as important to meaningful understanding as discursive reasoning and lower order logical reasoning. Russell tried to put logic on a more solid foundation but gave up. That may have been the right call. But I don't think it's irrelevant to consider that, whatever forces and organizational laws govern other physical/natural processes, our thought processes are constrained and influenced directly by the same factors, if any of them truly are universal in some real sense. It may be that by closely examining and thinking critically about human ideas, we're bumping up against and making inferences that really do relate to the fundamental structure of reality, but mixed in with that sort of "discovery" are various misleading ideas that aren't a product of indirectly/accidentally getting at those points of connection between human thought processes and natural reality but are themselves products of mental habits and errors that exist in the gaps in our thought processes that aren't strictly governed by mechanical, deterministic constraints.

In other words, maybe our cognitive processes are an imperfect reflection of the natural world, but they aren't ever really fully independent of the natural world, so even pure reasoning is to some extent shaped and influenced by natural law. The difficulty lies in correctly discerning the meaningful patterns from the operating noise of the machinery in the system. We've had a lot of success doing that for some reason. With all due respect to Hume, if that's only coincidental, it's a hell of a coincidence.
posted by saulgoodman at 6:40 AM on March 8, 2017

If the human animal had never existed would there still be math?
posted by judson at 7:17 AM on March 8, 2017

That's basically the same question as "Does a tree falling in the forest make a sound?" In that case, it depends on which definition of "sound" you use to parse the question: the one that describes a human sense experience or the one that describes the wave propagation of a particular band of electromagnetic frequencies. There is a necessary and direct relationship between both senses of the word, but they are describing different phenomena.

In the case of "math," likewise, it depends on how you define the term for purposes of analysis, whether you mean the language of mathematics, the conceptual objects of mathematical language, or the practice of doing math.

I'd argue some of the objects of mathematical language and reasoning do seem to exist independently of mathematical language and math as a human practice, as Goedel believed he had shown and arguably did.

Depending on what you're really asking, you get different answers. Bees and other species intuitively use complex mathematics to navigate and explore the world. In that sense, of course there would still be math if humans had never existed. There just might not be anybody around consciously noticing it and trying to figure out how it works to use it to make predictions.
posted by saulgoodman at 7:52 AM on March 8, 2017 [1 favorite]

...counting is something that happens in our minds. Planets are countable because we observe and count them, but it's a property that we impose on the planets and not one that they possess in and of themselves. Planets (used to) have mystical significance too, but I hope you wouldn't say that Venus is intrinsically female or that Jupiter has anything to with royalty.

But whether the mass that orbits the Sun at the relative location of Jupiter is called "Jupiter" or "Odin" or "The Great Gazoo" or "Bowman" does not affect the behavior of that phenomenon or future events.

Conversely, if there were two masses above a certain magnitude there, as there are two masses above the magnitude of the Moon at the Earth-Moon barycenter, that would produce a substantial difference in the behavior of the Solar System and in the course of future events; or if the characteristics of that mass were such that it would be described as a "star" rather than a "planet"; as would be the case if circles in general, or particular circles related to Jupiter, had a circumference-to-diameter ratio other than π, or if the shifts between electric and magnetic fields in the phenomenon we call "light" would be described with some sort of monotonic function rather than trigonometric functions.

If you had two different groups of humans who had never in history communicated with each other, keen-eyed and observant individuals among them might use different and completely unrelated names to describe the point of light that rises in the night sky where Jupiter is and moves across the field of stars over weeks and months, but we would expect that no observer to whom the question could be put would say that there are more than one of them.

So two-ness, and other mathematical means of description, at least seem to be connected to the real world in a different way than the names humans use for identifying things are.

You can change the way you mathematically describe something, for example speaking of four-sixths rather than two-thirds, but the different descriptions must be something like isomorphisms of each other to not cause difficulties in the description of the rest of the universe.

we define a set of things and that makes it countable

Incidentally not all sets are countable; this is what I meant by asking whether countable objects are uncountable when we're not looking. You're the one who put forward the premise that objects definitely exist; I'd probably go with something like maxima and minima for the purpose of this conversation, if I were going to talk about countable things.
posted by XMLicious at 4:30 PM on March 8, 2017

One of my English department colleagues gave his students an assignment a few years ago to interview other profs about their disciplines. In the assignment, he described academic disciplines as being defined by (1) the topic of study and (2) the methods of inquiry. I find this to be a useful starting point when talking about these sort of philosophical discussions, because my definition of math, as a mathematician, can differ quite a bit from what other folks think of math as being.

My description of the topic and methods of mathematical study would be pretty close to those outlined by Lockhart in "Measurement" (which is a truly excellent and approachable read for people of all mathematical ability and interest levels, on a side note):

(1) In math, we study "mathematical objects", which are completely abstract things that we have to define with rigorously precise definitions because they don't have physical existence per se - we can't bring a perfect circle into a lab to study it and to measure the ratio of the radius to the circumference, for example. Mathematicians have settled on the language of sets and relations to define mathematical objects, so one could say that math is the study of sets and relations on sets. But I think this idea of abstract objects given by our mathematical definitions, that Lockhart describes, is a more clear description.

(2) We study the properties of mathematical objects by the methods of logical deduction. More or less - that is, not all areas of math can be axiomatized (Russell and Whitehead tried, and failed). Another book I've read describes a mathematical proof as a convincing argument, with the caveat that mathematicians are trained to be very, very skeptical and to require very rigorous arguments in order to be convinced. This is a little closer to the truth of how math is actually practiced as an academic discipline for the areas of math that are harder to axiomatize.

Modern philosophers and sociologists of mathematics would probably add a third component to the definition of an academic discipline, to include the sets of factual propositions and values widely held by practitioners of the discipline (related to the idea of the social construction of knowledge). In math, this ties in to (2) and the issue of disciplinary standards for what counts as a valid mathematical proof.

With the above definition of mathematics, I think it's pretty clear that math is invented. But, contrary to popular myth, much of math is not invented purely on whim. Rather, there are community standards about what would make for a "reasonable" or "natural" definition of a new mathematical object, eg. based on past history and definitions of similar objects. And that past history (as well as a fair amount of modern but still technically "pure" math) is based on math that was developed specifically to model the real world. So: a perfect circle doesn't exist in the real world, that's an abstract concept that humans invented. But humans invented it based on or in reference to objects in the real world that our pattern-forming minds grouped together as similar. Our pattern-forming minds thought these objects were similar based on a "circle-ness" observed property, that we then abstracted into the mathematical definition of a circle. But, given that we lacked well-developed ideas or terminology about epistemology, cognition, and meta-cognition back in the day, we don't really have separate words to talk about the common property of the real world objects separately from our abstract concept of a circle. (I think that this conflation underlies a lot of the what-I-think-are-mistaken-claims of math being discovered.)
posted by eviemath at 6:36 PM on March 8, 2017 [6 favorites]

It may be that by closely examining and thinking critically about human ideas, we're bumping up against and making inferences that really do relate to the fundamental structure of reality

It's my considered opinion that "the fundamental structure of reality" is itself a deeply dubious thought-form; that structure in general and fundamental structure in particular are themselves our own conceptual constructs, and that there is plenty of room for a diversity of opinion about which kinds of structure matter.

I think there's a lot of overlap between the mathematical/logical idea of an axiom, and the idea from physics of a "fundamental law of Nature" and/or "theory of everything". In both cases one has something from which "all else follows".

Mathematics contains vast towers of propositions rigorously derived by applying deduction to axioms and its methods have certainly had a massive influence on physics. It's really tempting to look at something like the Standard Model, see it as akin to a collection of mathematical axioms, note that it supports a vast tower of successfully predictive physical theory, and conclude that the Standard Model is a description of "the" "fundamental" "structure" of reality.

I think every single one of those scare-quoted words bears close examination, in this context. Because it's in no way clear to me that reality itself is indeed "constructed", like a bunch of mathematical theorems, from anything "fundamental".

I think mathematics as a discipline and a method was completely unambiguously invented, and in no sense existed before we did that. I think that many hitherto-unexpected relationships between mathematical concepts have been noticed/discovered/intuited/demonstrated as opposed to invented, even though many of those discoveries would not have been possible without the invention of new mathematical methods.

But the almost completely local products of our own engineering aside, I think the universe is overwhelmingly indifferent to our inventions of both mathematics and physics. It will keep on doing as it damn well pleases^* regardless of whether we think it's "fundamentally" made of particles, or waves, or fields or strings or branes or consciousness or information or love.

^*Rhetoric inserted for illustrative purposes only. Actual universe may differ from that depicted.
posted by flabdablet at 7:40 PM on March 8, 2017 [2 favorites]

our own conceptual constructs, and that there is plenty of room for a diversity of opinion about which kinds of structure matter.

Don't absolutely agree with the first part, certainly agree with the spirit of the second part, but we might both be too physically limited to know what we're talking about, so I would never go to war over it.

"Fundamental" to me here just means something like baked into the structure of space/time, which definitely does have an observable structure.
posted by saulgoodman at 8:07 PM on March 8, 2017

it's in no way clear to me that reality itself is indeed "constructed", like a bunch of mathematical theorems, from anything "fundamental".

I'm not sure whether you're disputing the constructive nature of reality, or the idea that there is any fundamental level to these constructions - i.e., you think we may be living in a holograph made of waves made of branes made of whatever and so ad infinitum. I think they're both defensible positions, but the first one requires that you define "reality" as being something other than what we experience and can measure.
posted by Joe in Australia at 8:11 PM on March 8, 2017

I'm not sure whether you're disputing the constructive nature of reality, or the idea that there is any fundamental level to these constructions

Why not both?
posted by flabdablet at 8:12 PM on March 8, 2017

I think mathematics as a discipline and a method was completely unambiguously invented, and in no sense existed before we did that.

Is anyone actually arguing otherwise? Many of the comments in the thread seem to vigorously attack the notion, but I don't feel as though anyone here or in the OP video clips I've watched is trying to make the case that the academic discipline or the particular details of how humans go about examining mathematical questions have intangible existence apart from humans.

I think that the mathematical relationships humans have noticed might well just be very special cases of things that would be alien and incomprehensible to us if viewed in their entirety, and that the way we express them might well rarify and make tenuous their connection to anything real.

I just think that the fact that the things we perceive as mathematical relationships appear to constrain disparate events and behaviors of phenomena in what we would call the real world, even when we aren't examining or describing those events and phenomena, signifies that we are in some way perceiving things that are not confined to our own minds or human culture, that they aren't like lens flare, produced as artifacts of the instruments (human minds and culture) we're using to examine things.

We'll never know for sure as we can't escape the human condition, though it would be nice if we met extraterrestrial aliens at some point and they turned out to have some analog to mathematics and so we'd know that at least it would have to be an artifact of some category of thinking beings in general, but saying that mathematical relationships which appear to constrain everything everywhere we look at have no more reality than the names we give to planets seems bizarre to me.

I mean yeah, you can say that all of reality is constructed and therefore mathematics is constructed even if it has bearing on reality that transcends human context, but that seems like a tautological and rather unsatisfying analysis of the question.
posted by XMLicious at 8:19 PM on March 8, 2017

but the first one requires that you define "reality" as being something other than what we experience and can measure.

That's a good point. To some extent, that is the case, because we're stuck inside the bubble of our senses, which are mechanically connected to reality, but represent reality to us arbitrarily, using placeholder variables (qualia) for every mysterious piece of information we take in, like in algebra. Even the real numbers themselves are just a slightly more specific kind of variable, specifying only a generic count property that can be reused to describe that same property in any number of specific cases; it combines rigorous specificity with expansive generalization in one lexical unit. Math is powerful because its structures can be both very specific and very abstract at the same time, along different dimensions.
posted by saulgoodman at 8:22 PM on March 8, 2017

I don't feel as though anyone here or in the OP video clips I've watched is trying to make the case that the academic discipline or the particular details of how humans go about examining mathematical questions have intangible existence apart from humans.

Stephen Hawking: "Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?"

I've always read that "breathes fire" question as containing an implicit assumption that the equations of a correct unified theory must have some kind of existence that's independent of the universe they describe. There are many other authors (Paul Davies, for one) who also seem to be coming at the thing from that angle.
posted by flabdablet at 11:20 PM on March 8, 2017

Well, sure, from that one sentence that's the implication of the cosmology he's entertaining the idea of. I don't see how it's an argument that the human discipline and method of mathematics is not invented, though.

If we humans correctly ascertained the way to express some rule integral to the operation of the universe in our symbolic representation of mathematical rules, if that were the way the universe works, it would not mean that the method we used to reach the conclusion or the academic discipline which studied it had been discovered rather than invented because they'd existed before humanity ever did.

Maybe I misunderstood the sentence of yours about "mathematics as a discipline and a method" I quoted above? The equations describing a correct unified theory would seem to fit into the subsequent sentence in your original comment concerning things which have "been noticed/discovered/intuited/demonstrated as opposed to invented".
posted by XMLicious at 11:49 PM on March 8, 2017

I don't see how it's an argument that the human discipline and method of mathematics is not invented, though.

Agreed.

However, there is a fairly widely held opinion that numbers, sets, ideal geometric forms and so forth really do have some kind of existence independent of the human discipline and methods we've developed to describe, categorise and investigate them, and could fairly be considered real things even if we'd never actually developed any such discipline.

It's an opinion with which I strongly disagree, but there's no doubt that it's out there.
posted by flabdablet at 12:09 AM on March 9, 2017 [1 favorite]

What would you mean by saying that numbers aren't real things, though? They're more real in some way than things like the names we give planets, right?

The names of the numbers don't have existence apart from humans, nor aspects like representing them in decimal rather than binary, etc., but they seem to have some kind of existence insofar as material things behave in predictable ways based upon inherent properties that correspond in some way to numbers we would use to describe them.

"More than" and "less than" seem to have some relation to reality; if you sit on one end of a seesaw that has a small child on it, the end you sit on will sink down no matter how fervently you deny the reality that your "weight" is "more than" that of the child.

Or, a hydrogen atom that has "one" neutron in its nucleus has different properties than a hydrogen atom with "two" neutrons.

On the up side, if numbers are not real, we all get to use the royal "We" without being queens or kings, because it can't be validly said that any of us are individuals. At least for anyone who is real, that is; people who aren't real, being in the same realm of unreality as numbers, can perhaps have a number assigned to their quantity of selves.
posted by XMLicious at 1:32 AM on March 9, 2017 [2 favorites]

they seem to have some kind of existence insofar as material things behave in predictable ways

But who is doing the predicting?

We are.

And we invented numbers to help us do that. Numbers exist inside the minds of entities like us who value predicting things. No such entities? No numbers either.
posted by flabdablet at 1:49 AM on March 9, 2017

Using the same logic, anything we think about ends up existing inside our minds, so nothing anyone has ever thought about should exist.

Or, if making predictions is the pivotal thing, predictions can be made with things other than numbers. I predict that you will make another comment. But that doesn't mean that you don't exist, or that comments don't exist.
posted by XMLicious at 2:08 AM on March 9, 2017 [1 favorite]

anything we think about ends up existing inside our minds, so nothing anyone has ever thought about should exist.

I can't see how you get from making a mental representation of some part of reality to rendering that part ipso facto nonexistent. If I observe three elephants, and I say to myself "there are three elephants over there", the elephants don't stop existing.

However, the fact that there are three of them exists only as an idea in my mind; it's not inherent in the elephants themselves.

The three-ness I observe relies completely on my implicit categorization of that observed phenomenon and that observed phenomenon and that observed phenomenon as examples of the category "elephant", any my ignoring every feature of those phenomena except that categorization.

If you're nearby, and you're also looking at the same elephants, it's entirely likely that you will also have an experience of three-ness. However, to my way of thinking that's not because three-ness is inherent in the party of elephants; it's because both you and I are both representation and categorization engines, and we're both wired up in pretty similar ways.

Alternatively, you might be standing in a slightly different spot from me, from which you can see that the group actually consists of four elephants.
posted by flabdablet at 3:11 AM on March 9, 2017 [1 favorite]

But... there are either actually three elephants or four elephants. It's not just in your mind. For example if the elephants walk over an area with four pits in the ground, and afterwards there is an elephant in each pit at the same time, then there weren't three elephants. The four-ness of the elephants and the four-ness of the pits exist in reality and relate to each other in a way that constrains possible future events, whether there are any humans there to observe it or not.

Yeah, someone can be mistaken about the number of elephants, or could mistakenly think that they're something other than elephants, but the very fact that reality can assert itself over that misunderstanding—if for example the observer shoots three elephants to death and is subsequently trampled by a live uninjured elephant, or if the observer tries to drive his ATV over four small grey hills and the hills stand up and run away from him—would seem to confirm to me that both the fact that they're elephants and the fact that there are four of them are aspects of reality, not things which solely exist in the observer's mind.
posted by XMLicious at 5:01 AM on March 9, 2017 [1 favorite]

What would you mean by saying that numbers aren't real things, though?

I think for some people, the notion of any non-physical thing being real is on the suspicious-to-unacceptable-to-nonsense spectrum
posted by thelonius at 5:03 AM on March 9, 2017 [2 favorites]

Maybe the thing is to think of it as a system of elephants with cardinality four. A loosely-bound quadraphanton.
posted by XMLicious at 5:15 AM on March 9, 2017 [1 favorite]

I think for some people, the notion of any non-physical thing being real is on the suspicious-to-unacceptable-to-nonsense spectrum

...Which is a form of known cognitive impairment, incidentally... Inability to form and intuit abstract concepts is associated with cognitive development problems. If something about our society/culture is reifying and reinforcing basic cognitive malfunction as ideology or as a positive cultural value, that might not be a good thing. In fact, overemphasis on physicality and anti-intellectual bias against narrative and complex abstract thought and idealism have long been recognized as markers of protofascist societies. The fact some of these unscientifically reductive, strictly materialistic theories of reality are embraced on both the political right and left worries me. Naive physicalism isn't anymore scientifically sound than new age quantum woo; reality is still more complicated than that, and it turns out that even the observable solidity of "matter" is basically due to particular electromagnetic wave interference patterns. In classical Taoist thought, they might have argued this was intuitively obvious, and that it makes perfect sense matter itself has to be composed of constituents that aren't themselves whatever we mean by the term "matter," and what do you know, that's exactly what we observe in nature, where matter appears to be more of an epiphenomenon, an emergent feature of interacting forces that aren't themselves material in the naive, atomic, epicurean sense, though you might fairly describe them as metaphysical or natural in the sense of "natural sciences."
posted by saulgoodman at 8:26 AM on March 9, 2017

I'm sure that no one here has a problem with non-physical things being real in general, though. No one is going to shout "Witchcraft!" if they see iron filings scattered around a bar magnet align into lines of force or Mythbusters snuff out a candle with a speaker and a tone generator. It's just the ages-old open question of what mathematical things are real independently of human invention, and to what degree.
posted by XMLicious at 10:25 AM on March 9, 2017 [1 favorite]

Maybe not here, but believe it or not, there are still even die-hard epicureans out there, lol
posted by saulgoodman at 12:46 PM on March 9, 2017

But... there are either actually three elephants or four elephants. It's not just in your mind.

An observer with a different cultural background might say that there are two cow-elephants and one bull-elephant, and they're obviously different classes of things. Or they might say that there is one group-of-elephants, just as a flock-of-sheep or swarm-of-bees is a single thing. A hypothetical field-based lifeform might be confused about the idea that you can meaningfully distinguish between interacting globs of matter, and a one with a different understanding of time might struggle with the idea of an individual elephant, rather than a network of past and future elephants.

If those distinctions seem unrealistic, imagine a knight who boasts that he fights for three things: for his faith, his comrades, and his king. That's a genuine enumeration: we can say that three is a smaller quantity than the four things that another knight fights for, but greater than it would be if the knight "lost his faith" and only fought for two things. But I think it's weird to say that these three things have numerical attributes in and of themselves: even kings are a matter of consensus, not natural law. At some point you're reduced to saying "if we count James III as a king, and a belief in natural justice as a faith ..." and at that point you're clearly assigning numerical values to things on an arbitrary basis. The things aren't necessarily in your head, but that's where the counting and countability reside.
posted by Joe in Australia at 1:46 PM on March 9, 2017 [2 favorites]

So if somebody built a machine for counting elephants, it wouldn't work when no one was looking or in places where people believe different things?
posted by saulgoodman at 2:09 PM on March 9, 2017

That's Searle's Chinese Room argument in a different form. If you can build a machine for counting elephants then it will use the definitions you gave it ("a mammoth would be an elephant, but no dead creature is an elephant; a picture of an elephant is not an elephant; an elephant with a person riding on it is still an elephant ...") and it will be designed to give you an answer that's consistent with your expectations of what is "countable". If the machine was designed by hypothetical aliens it would give a different answer, possibly counting things that you don't even imagine to be countable, e.g.:
1) An ignored problem ("an elephant in the room") is an elephant;
2) The ("elephantine") motion of a bulky vessel or vehicle makes it an elephant;
3) A neighbourhood in Brooklyn ("Dumbo") is an elephant ...

Surely you don't think that the physical natures of "ignored problems", "heavy motions", neighbourhood nicknames, and pachyderms, are all sufficiently similar as to be capable of possessing an attribute in themselves. But they're all countable, so that attribute must exist somewhere. I say that it's in the mind of the person doing the counting.
posted by Joe in Australia at 2:38 PM on March 9, 2017 [2 favorites]

We don't have to treat this one as just a thought experiment necessarily. You could build a machine that accurately counts elephants right now. That's not beyond current technological capability. Do you really think what people believe would affect its function specifically in this case? You have a point about culture being very important, but there's no guarantee any particular cultural value or belief is compatible with scientific knowledge. And I have seen no evidence in 43 years that belief and culture have any fundamental effect on the laws of physics or the function of natural processes. It's only in interpretations of more complex problems that culture becomes confounding. Simpler example, if I built a machine to count golf balls from a bin in a box attached to it, is there any scenario where belief changes anything about the count?

Could you swap out the golf balls with the same number of different colored balls and get the same count? Why if there's not something really in common in natural reality between the two batches of balls?
posted by saulgoodman at 3:32 PM on March 9, 2017

If one of the golf balls breaks in half, how many are there now? Will every copy of the same golf ball counting machine design give the same answer?

If one of the visible elephants is pregnant, are there still four? Or if one of the elephants is in the process of dying, when are there no longer four elephants?

I think the label four, just like the label elephant, is necessarily subjective for everyday objects, even if most people would agree on the label most of the time. (Which doesn't mean at all that there isn't a shared physical reality.)
posted by mubba at 6:45 PM on March 9, 2017

You could build a machine that accurately counts elephants right now.

Only if you can define what an elephant is – and I will stipulate that it can be done with sufficient accuracy to satisfy most people. But we've already got machines that count things like days of the week, or gross national product, or "pages that have been read". Do you think there is an inherent "countability" to those things in the absence of human observers, and if not, how does their countability differ from the countability of elephants?
posted by Joe in Australia at 6:54 PM on March 9, 2017 [2 favorites]

Which doesn't mean at all that there isn't a shared physical reality

Exactly. All it means is that talking about that shared physical reality requires representing/modelling it; from which it follows that what's generally thought of as an objective description of physical reality is actually a description of a consensus-based model.

For most purposes this is a distinction without a difference.

For the purpose of considering issues hinging on the operation of the representation engine itself - such as whether mathematics (and by extension physics) is an invention or a discovery - it isn't.
posted by flabdablet at 7:11 PM on March 9, 2017

In other words: when Penrose describes mathematics as "something out there that seems to have a reality independent of the ordinary kind of reality, like things like chairs and so on which are what we normally think of as real", I think he's mostly right; the only part I'd take issue with is that instead of "out there", I'd say "in here".
posted by flabdablet at 7:23 PM on March 9, 2017

And of course I fundamentally disagree with "mathematical facts, like 'there is no largest prime number' - it's something independent of ourselves, it's always been true".

No representation engines? No numbers. No numbers? No prime numbers. No prime numbers? No facts about prime numbers.

Even more fundamentally: No representation engines? No truths.

To claim that "there is no largest prime number" has always been true is, in my view, a prime example of mistaking the map for the territory.
posted by flabdablet at 7:32 PM on March 9, 2017

The fact that all of these religious arguments rest squarely on that eternally useful handwave, "in a sense", should tell us something as well.

In whose sense?
posted by flabdablet at 7:37 PM on March 9, 2017

>> But... there are either actually three elephants or four elephants. It's not just in your mind.

JiA > An observer with a different cultural background might say that there are two cow-elephants and one bull-elephant, and they're obviously different classes of things. Or they might say that there is one group-of-elephants, just as a flock-of-sheep or swarm-of-bees is a single thing. A hypothetical field-based lifeform might be confused about the idea...

This is one reason why I mentioned Heraclitus and panta rhei above: precisely because specifying arrangements of matter and other phenomena is arbitrary, you might as well say that perfect circles exist as well as you would say that any object exists at all.

Note that I also anticipated your "group-of-elephants" proposal, which I phrased as "a system of elephants with cardinality four."

Just like with construing an observer as having been mistaken by what they're seeing, the very fact that you have to start fiddling with definitions to overcome the fundamental difference between three-ness and four-ness to, for example, get three things to fit into four pits, again confirms for me that numbers in some sense have connection to analogous things in reality and produce constraints within that reality. Like, one of the things could also travel back in time and simultaneously be in two of the pits.

Do positive integers exist but not all integers and not real numbers? Or natural numbers but not irrational numbers? Are the things that are real actually almost completely unlike what we think of as numbers, or are numbers a very special case of something else? I don't know, but if one is going to say that anything at all is real I haven't come across a version of that notion yet which appears to be able to also deny the reality of all mathematical things.

flabdablet > Even more fundamentally: No representation engines? No truths.

As I said above (in the final paragraph), you can certainly just claim that nothing is real and definitively prove that as a corollary, no part of mathematics is real. It's just that you haven't said much, if under the claims of things not being real that's all there is.

It's also kind of odd that you have said this after previously indignantly denying that your reasoning leads to nothing at all existing, when the question was specifically put to you. I'd have thought you'd at least mention "Well yes, nothing exists, but for different reasons."
posted by XMLicious at 11:56 PM on March 9, 2017

Note by the way that even solipsism is impaired by denying the reality of numbers: you can't posit "I think therefore I am" as the fundamental truth of the universe if you have no way of claiming there's only one "you".
posted by XMLicious at 12:18 AM on March 10, 2017

"natural numbers but not irrational numbers" should actually be "rational numbers but not irrational numbers" above
posted by XMLicious at 1:21 AM on March 10, 2017

you can certainly just claim that nothing is real

I suppose I could. I've specifically and repeatedly not done that, and it's not a position I agree with, but if you wish to believe that that's what I think, it's no skin off my nose.
posted by flabdablet at 2:48 AM on March 10, 2017

So things can be real but not true, in your framework? I mean, you specifically said that there are no truths in the absence of representation engines.

Basically, what can be said to be real in the "shared physical reality" you mention, if nothing is true? Are the elephants real?

You appear to heavily emphasize the existence of numbers within the representation engines, as if their presence there somehow demonstrates that they aren't real. But I am missing why this doesn't apply to the other things that appear in the representation engines, or how you're doing more than simply arbitrarily asserting that out of all the things which appear in the representation engines, numbers and mathematical things are the ones which aren't real.

If you're saying that the elephants are real and numbers aren't, why is that the case rather than the numbers being real and the elephants not?

I guess I should also ask if "discovered rather than invented" is synonymous with "real" or "true", or if there are three separate terms here rather than different ways of speaking about the same thing, as I've been intending to use them.
posted by XMLicious at 3:10 AM on March 10, 2017 [1 favorite]

To relate this a little bit more directly to one of your previous statements:

The three-ness I observe relies completely on my implicit categorization of that observed phenomenon and that observed phenomenon and that observed phenomenon as examples of the category "elephant", any my ignoring every feature of those phenomena except that categorization.

It seems to me there's no reason the category must be "elephant" rather than the categories being "quartet" and "triple" and elephant being the non-real aspect or facet of items in those categories, "quartet of elephants" or "triple of elephants". If you're saying that some things which appear to be real are merely artifacts of the internal operation of representation engines, I'd think you would still need to explain how the real things are distinguished from the non-real ones.
posted by XMLicious at 3:43 AM on March 10, 2017

you specifically said that there are no truths in the absence of representation engines.

Truths are statements about things. Statements about things are representations of things. Regions of reality that contain no agents capable of making representations (such as the planet Earth before the advent of sentient life) therefore contained/contain/will contain no truths.

This is not to say that you and I (who are, I hope uncontroversially, representation engines) are unable to generate truths about those regions. We can, and demonstrably do. Those truths (i.e. those accurate representations of regions of the shared reality we all inhabit) are real, as are all the representations we make and use, and they exist within our brains. We do not fully understand the form in which they exist, but we have enough decent clues to have engineered neural networks that we can be fairly confident operate in broadly similar ways.

The edifice of mathematics is a complex and interlocking collection of truths. All mathematical objects are, therefore, representations. Many of the things that mathematical objects represent are other mathematical objects, which can in and of themselves consist of relationships between yet more mathematical objects. All of these representations are real, and they exist within our brains.

Physics is also a complex and interlocking collection of truths. Much of it overlaps with mathematics. Compared to mathematics, though, more of the truths of physics are about those regions and aspects of shared reality not physically enclosed by our skulls.

You appear to heavily emphasize the existence of numbers within the representation engines, as if their presence there somehow demonstrates that they aren't real.

Sure they're real. They're just not external to the representation engines. They are working, functional parts of those engines.

I guess I should also ask if "discovered rather than invented" is synonymous with "real" or "true"

The way I see it, invention requires creativity and a tremendous amount of work, while discovery merely requires a tremendous amount of work.

We can discover truths about our shared reality. We can also invent new ways to go about performing that discovery.

Does any of that help?
posted by flabdablet at 3:44 AM on March 10, 2017

Sure they're real. They're just not external to the representation engines. They are working, functional parts of those engines.

But how do we know that? What's the evidence that they are real-but-internal rather than real-and-external?

Without addressing that you're effectively just re-stating the premises of the question about whether mathematics is discovered or invented, rather than trying to answer it.
posted by XMLicious at 3:54 AM on March 10, 2017

If you're saying that some things which appear to be real are merely artifacts of the internal operation of representation engines

I very very rarely use the word "merely", and even more rarely take it seriously. And I have not claimed, and would not claimed, that artifacts of the internal operation of representation engines are not real.

What I do claim is that those artifacts can, and in the context of discussions like the present one should, be considered as non-identical with those aspects, regions or features of our shared reality that they represent.

This is kind of slippery to think about, because "aspects, regions, features" are themselves artifacts of the internal operation of representation engines i.e. us.

Key thing to hold onto is that the fundamental operation of reason is the distinction - the choosing of some way to divide thing from not-thing.

In order to talk about elephants, we need to make a distinction between that elephant-shaped chunk of reality over there and the negelephant that surrounds it. And as soon as we've done that, we're talking about our models - i.e. the ways we perceive reality - and not about our shared physical reality itself.

In fact the only thing we can actually do with our shared physical reality directly, without making and using representations of it, is experience it.
posted by flabdablet at 3:57 AM on March 10, 2017

But how do we know that?

Reductio ad absurdum works for me. Assuming the opposite ends up with the word "exist" acquiring contradictory meanings that can only be sustained by utterly absurd amounts of handwaving.
posted by flabdablet at 4:01 AM on March 10, 2017

That sounds rather like "just because" to me... I feel as though you're almost literally saying "obviously I'm right that numbers have only real-but-internal existence because anything besides what I'm asserting would be absurd." I feel as though I've repeatedly demonstrated that numbers have real-and-external existence, or some connection, however tenuous, to things which have real-and-external existence, in some manner which affects other things in the external world, regardless of how any human chooses to think about it or not think about it at all.

I guess it seems self-evident to me that the question is not at all whether mathematics exist as thoughts in the real heads of real humans, but thank you for affirming that. "Merely" is certainly my word, attempting to distinguish the manner of existence which seems relevant to the question from the manner of existence that does not appear to me to bear on the discussion very much, two things which seem to be getting entangled, from my perspective.
posted by XMLicious at 4:19 AM on March 10, 2017

i.e. the ways we perceive reality - and not about our shared physical reality itself.

thinking of things as "physical" is a "model" too btw
posted by thelonius at 4:28 AM on March 10, 2017 [1 favorite]

I feel as though you're almost literally saying "obviously I'm right..."

Again, close but not quite. I'm saying "this stuff seems fairly obvious to me. Your mileage may vary".

I dropped the "I think..." and "It's my opinion that..." qualifiers once you and I started having a bit of a back-and-forth exchange of views, because I thought it would have been pretty obvious by then that those qualifiers are implicit.

All of us represent the world we inhabit in different ways - with some gross differences and some quite subtle - and that's usually perfectly OK. It's also pretty good evidence for the idea that our brains are indeed representation engines with our life experiences as training data; I would not expect your representation of the world to match mine in all respects.

thinking of things as "physical" is a "model" too btw

Agreed. Thinking about anything necessarily involves modelling it.
posted by flabdablet at 4:35 AM on March 10, 2017

Yes, humans exist and think things in their actual real brains and do so in a different fashion from one another, (if indeed there can be said to be one human or more than one human, somehow, even though numbers do not have external existence) but this continues to seem to me like intricately laying out the premises of the question without addressing it.

If the number of elephants, or planets, or neutrons, or maxima and minima of fields in any given situation, seems to have a material effect on the state of the external world from one moment to the next, and to constrain what subsequent events are possible, and appears to produce congruences, so that four or more elephants can simultaneously fall into four pits but three cannot, no matter whether or not humans are observing events or indeed exist anywhere, why would that be the case if numbers have no existence whatsoever in the external world?

(And of course, elephants and pits can be replaced with an example involving planets, neutrons, or maxima and minima, or something else if you wish—you brought them up, but I like elephants.)

(Furthermore, we can run through a figuratively infinite number—ha—of specifications scoping out and fine-tuning the particulars of any hypothetical scenario I might present where a number appears to matter, and similarly go on and on theorizing about how it would be represented in a human mind, but these activities seem to me just a way to postpone tackling the actually-salient issue of why the characteristics of the scenario in question which can be described with numbers appear be characteristics which would have a material effect on the external universe. Unlike, say, the name we might give an elephant or a planet.)
posted by XMLicious at 6:13 AM on March 10, 2017

That last bit might be more clearly worded: ...why the characteristics, which can be described with numbers, of the scenario in question appear be characteristics which would have a material effect on the external universe.

(Talking about "characteristics which can be described with numbers" rather than numbers themselves in the most thorough formulation of the question because, as I've noted before, perhaps the externally-existent things which numbers correspond to are very different from numbers themselves.)
posted by XMLicious at 6:28 AM on March 10, 2017

If the number of elephants, or planets, or neutrons, or maxima and minima of fields in any given situation, seems to have a material effect on the state of the external world from one moment to the next, and to constrain what subsequent events are possible, and appears to produce congruences, so that four or more elephants can simultaneously fall into four pits but three cannot, no matter whether or not humans are observing events or indeed exist anywhere, why would that be the case if numbers have no existence whatsoever in the external world?

I'm in no way attempting to deny that numbers (and mathematics generally) are useful components of models we can use for making successful predictions about the behaviour of certain interesting parts of reality. It would have been pointless for us to have invented them otherwise.

Could you perhaps fill in a few of the steps between our ability to count elephants and count pits, and numbers existing in the external world? I'm not seeing how that follows.
posted by flabdablet at 7:35 AM on March 10, 2017

Well, the thing is that from my perspective we don't have to count them or even be able to count them. If a group of elephants with three-ness walks over a group of pits with four-ness, there is no possibility of an outcome where each and every pit contains an elephant, whether or not humans even exist.

Thereby, in the same way that other characteristics of the group of elephants which we might regard as real (in that they constrain possible future events in the external world in a given set of circumstances)—has "relative location" within the Earth's rest frame which governs other objects and phenomena it might interact with, has "mass", so it won't be departing its current location at the speed of light^†, has a "temperature" in a range such that the group of elephants will not behave like plasma—in the same way, the three-ness constrains future events in the external world.

If you accept the above characteristics I've described as externally-real, that is.

Location in particular seems like a good analogy to what I'm trying to propose about numbers being connected to something within external reality: with special relativity, general relativity, quantum phenomena, and who-knows-what we have yet to discover, even though it seems to be in some way real, the reality is bizarrely and convolutedly distant from the sort of thing we might intuitively regard to be location.

I of course don't regard my description of our group of elephants to be in any way canonical. JiA's hypothetical field-based life form would probably look at all of these characteristics differently, in an equally valid way. I'd think that the three things we're calling elephants would be the local maxima of some sort of composite field that would be useful to the alien in the same way our mental models are useful to us, and the fact that there were three of them would have the same significance regarding possible future events as us saying "there are three elephants" does.

† Elephants are still elephants in the dark, so reflected light is of no consequence, but does the infrared radiation they emit count as part of them? Presumably not, in the way that their farts do not remain part of them.
posted by XMLicious at 9:28 AM on March 10, 2017 [1 favorite]

the three-ness constrains future events in the external world

If by "constrains" you mean "reduces the probability of the occurrence of" then I'd agree with you, and go on to point out that probability is one of the mathematical techniques that we employ to predict stuff; in other words, to make our models fit reality better.

But future events will be whatever they will be, regardless of what we predict. This is why in theory there's no difference between "in theory" and "in practice" while in practice there is.

If we recognize three-ness here and four-ness there and make predictions on that basis, we can indeed be pretty confident that those predictions will subsequently be confirmed by experience. However, those portions of reality outside our control remain completely indifferent to our predictions. No amount of confidence on our part about three-ness and four-ness would prevent, for example, all three elephants and all four pits collapsing into a single huge sinkhole as soon as the first elephant falls into the first pit.

Thanks for this discussion, by the way. I'm learning a lot about the fascinating texture of those elephants' ears from you. Hope you're not offended by my continued insistence on being more interested in the way their tree-like legs hold them up.
posted by flabdablet at 3:38 AM on March 11, 2017

But future events will be whatever they will be, regardless of what we predict.

This is entirely true, and because of it I don't understand why you keep returning to talk about predictions. Exactly because the fact of there being three or four or some other specific number of things outside of anyone's mind, whether we would call them maxima or minima or objects or whatever, is what provides the actual constraint on future events, that is what offers evidence that numbers are an external feature of reality.

Even for parts of reality "within our control", if we make predictions with numbers that do not correspond to the numbers in the actual external situation, our predictions do not matter a whit to future events, because it's the actual numbers of elephants and pits in the external world which determines future events—not the numbers which any human used to make predictions.

Furthermore, this also happens if there is an observer, but that observer doesn't have what you're calling these models in their representation engines—if for example a baby who has only just developed object permanence and has no concept of counting or numbers is watching a group of elephants walk over a group of pits, that does not make it possible for three elephants to fall into four pits. The outcomes are still the same as if no one was observing at all.

No amount of confidence on our part about three-ness and four-ness would prevent, for example, all three elephants and all four pits collapsing into a single huge sinkhole as soon as the first elephant falls into the first pit.

And of course nothing would prevent a meteor from landing on the elephants and the pits, turning the elephants to vapor. But somehow fiddling with the details of the scenario in that way doesn't definitively demonstrate that elephants aren't real, I'm going to guess?

Do you see how your responses to this scenario where you only change the internally-existent thoughts of the observers, like having two observers perceive a different number of elephants, has no effect on the outcome, and it's only when you change the premises of the scenario in external reality, and convert four pits into one the way I converted the elephants to vapor, that it makes any difference? You have to change the external reality of the scenario to get different numbers in the outcome of external events because the numbers are external to any human mind that might be involved.

Of course the numbers of elephants and pits are not immutable, any more than their location, mass, and temperature are immutable.

Though, I note that you refrained from answering the question of whether location, mass, and temperature are externally-real, and unless I'm missing it, despite your comment about distinctions and elephants and negelephants above you didn't directly state that elephants are real.

I guess you've stated that human minds are real, but we still have the unresolved mystery of why then it's possible to speak of there being any particular number of them? So that you would not expect my representation of the world to match yours in all respects, as though it were possible to say that there were two minds or two representation engines between us. Maybe that's how telepathy works: the number "one" is real, but no other numbers, so we're all just the same mind.

Despite an unspecified reductio ad absurdum supposedly being the reason why numbers can't have any external existence, I feel like we're coming across an extraordinary amount of absurdum entertaining the possibility of that being true.

In any case—far from being offended, I really appreciate you having engaged with me and having put up with my prodding about the course of the discussion. I've always wanted to debate the discovery-vs-invention issue on numbers in particular and this has confirmed many of my expectations about how it would go.
posted by XMLicious at 6:02 AM on March 11, 2017

the actual constraint on future events

isn't actually there. There are multiple constraints on any reasonable prediction about future events, but we can't know what is going to happen beforehand; not with complete certainty. Not even Laplace's Demon could pull that off.

our predictions do not matter a whit to future events, because it's the actual numbers of elephants and pits in the external world which determines future events

Don't believe so. The only thing that determines what will happen is what actually ends up happening. When what happens is what we expect will happen, that speaks to the quality of our representation/model of reality; causality, like numbers, is something I understand to be a mental tool and map-reading aid and not necessarily a feature of every part of the territory.

fiddling with the details of the scenario in that way doesn't definitively demonstrate that elephants aren't real, I'm going to guess?

The elephants you and I are discussing right now are certainly not real: these are hypothetical elephants, strictly map features, and they are located within our skulls. Mine are dayglo orange with high-vis reflective stripes. What colour are yours?

I note that you refrained from answering the question of whether location, mass, and temperature are externally-real

All map features, in my view, and all very useful for describing and labelling parts of the territory in order to make predictions about how they will behave.

you didn't directly state that elephants are real

Let me do that now, then: elephants exist. There are real elephants.

I guess you've stated that human minds are real, but we still have the unresolved mystery of why then it's possible to speak of there being any particular number of them?

Numbers would be rather useless if we couldn't use them to count real things. Also, I'm not sure why insisting that numbers must have some kind of existence other than as activation patterns within our brains is useful.

Also not sure why you keep implying that activation patterns within our brains are not, in and of themselves, real. Sure, some of them represent things that are unlikely to be real, like high-vis elephants and Donald Trump's hair; that doesn't make the activation patterns themselves non-real.

I feel like we're coming across an extraordinary amount of absurdum entertaining the possibility of that being true.

That's mainly because we are, to some extent, talking past each other due to fundamental incompatibilities between our worldviews. I suspect that you're a categorizer and I'm a tagger.
posted by flabdablet at 7:35 AM on March 11, 2017

All map features, in my view, and all very useful for describing and labelling parts of the territory in order to make predictions about how they will behave.

Does this mean that relative location, mass, and temperature are internally-existent in human minds or externally-existent? Are you saying that elephants with location, mass, and temperature exist externally or only massless elephants that aren't anywhere and do not interact with other things by transferring heat, exist externally?

That's not quite right the way I should phrase those things being non-externally-existent but it seems as though possessing characteristics that are in some way like relative location, mass, and temperature would be somewhat inherent in the definition of those definitely-externally-existent elephants, independent of humans existing or making any predictions. (I again do not understand how we keep getting back to humans making predictions.)

Also, I'm not sure why insisting that numbers must have some kind of existence other than as activation patterns within our brains is useful.

It has nothing to do with whether they're useful or not, though—elephants do not exist externally because it is useful to us for them to exist, they simply exist, and thereby have concrete and objective effects on external reality.

The elephants you and I are discussing right now are certainly not real: these are hypothetical elephants, strictly map features, and they are located within our skulls. Mine are dayglo orange with high-vis reflective stripes. What colour are yours?

If you believe that us physically getting together, going somewhere that elephants live, and experimenting with real live elephants and pits might result in three elephants falling into four pits simultaneously, by all means say so because that would be fascinating to see. No elephants were harmed in the making of this internet discussion.

I feel like there's a thread running among your objections that amounts to something like "But humans aren't omniscient!" but I'm unable to determine how this would have bearing on anything. If I'm able to describe a scenario in which the number of elephants and the number of pits would have a material effect on the subsequent outcome, a scenario which because we can presumably agree it is sufficiently plausible and in no way inherently impossible may indeed even have already occurred in the history of the universe, I don't understand why it matters whether there also may have been or could be a similar scenario, but a sinkhole opened up, or a similar scenario, but human observers were present and unsuccessfully weighed probabilities and unexpectedly the elephants were extraordinarily sure-footed and didn't fall in, or a nearby Mt.-St.-Helens-like volcano exploded and killed everything for miles around before the elephants got close enough to fall into any pits, or the human observers were there but blind and did the blind men and an elephant routine and hence didn't successfully predict the outcome because they didn't even know there were any elephants or pits involved, even after all the elephants fell in pits.

The reason I'm offering that scenario is because I'd expect that we could agree it's an accurate representation of how reality works in general, because it could actually happen and the outcome would in general be materially different with varying numbers of elephants and pits, not because all other variations are impossible or anything to do with the capabilities of some type of human witness to predict the outcome.

Also not sure why you keep implying that activation patterns within our brains are not, in and of themselves, real.

I'm not insisting that. Like I said, "humans exist and think things in their actual real brains and do so in a different fashion from one another". I mean, I would think that I can't even express anything here that did not already at least exist as a real activation pattern in my own real brain. (At least I think my brain is real.) I thought we'd hashed it out that I'm distinguishing between things which internally exist within human brains during the portion of the universe's history while human brains exist, and things which exist externally and independently of human brains such that if humans had never existed said thing would still exist. I could try to insert the latter preceding phrase parenthetically every time I say "real" or "exists" but it seems like it would just make things difficult to read.

If I'm not understanding something properly, could you perhaps point out an example where I appear to be stating that something does not at least even exist internally as an activation pattern within a human brain? (if not both internally and externally, or externally-only, which I think covers all the combinations of ways we've been talking about things existing)

I mean, I assume that when we say that elephants are real, we aren't specifying that the word "elephant" exists in English with its grammatical variations in human minds, but that the type of thing which the English word "elephant" refers to in the external world exists, independently of English itself such that there were elephants long before English existed. (And if the world is lucky, elephants will also still exist when English no longer does within any human minds.)

That, and furthermore to say that elephants exist externally and independently of any human mind thinking in any human language, and concomitantly with the entire point of the discussion to say that elephants are discovered rather than invented, is what it means to say that elephants are real, yes?

—
I think it's not really an incompatibility between our worldviews—I feel as though I can ably switch between thinking in terms of categories, or tags, or fields, or a variety of other things—I think that the root of our apparent disagreement is that once one has asserted that almost any particular thing that can be briefly expressed in vernacular English is "real" or "discovered", like elephants or human brains or possibly even just thoughts in human brains, one has already engaged in a sufficient level of abstraction that it is incompatible, or at least demonstrably inconsistent, to deny that numbers are an abstraction that is real or discovered in the same way.

That's my intuitive feeling for it, and thanks again for putting this cluster of hypotheses to the test with me.

(I guess I should mention that one thing which does seem like it could potentially unravel the things I've tried to establish thus far, is where you appear to say that causality isn't real. Like, if an elephant, by existing, places no constraints on future events because the elephant could transform into a bowl of petunias in a spontaneous uncaused fashion, or anything else could happen, I'd certainly have to at least reformulate my arguments about effects upon the external world that demonstrate the reality of numbers. I would think causality not being real would have to be true in a fairly weird sense to only pose an obstacle to numbers existing but not elephants, but it might be worth exploring.)
posted by XMLicious at 11:12 AM on March 11, 2017

Discovery: there's a physical object that I can go observe. Exactly how I describe my observations will depend on my representational structure. Someone else might go observe the thing and describe it differently because they have a different representational structure. But we're both observing the same object. I make representational choices, but my representational choices don't affect the original object, and so don't affect an independent observer making their own representational choices (in the absence of any knowledge about my observations and representational choices).

Invention: I make choices that determine the actual attributes of the object. Generally I work within some system of rules (eg. the constraints of the physical universe, when inventing a physical contraption, or culturally defined tonal and lyric structures when inventing a piece of music), so it's not that I'm making arbitrary choices at every step of the invention process. But if a different inventor followed the same system of rules but made different initial choices, they would end up with a different invention, not just a different description of the same invention. Unlike with discovery, different choices that I make affect the actual composition of the invention, which can be observed by independent observers.

Some mathematicians really and truly do seem to think that math is discovered in this sense - that they make choices about what words to use for terminology or what symbols to use for notation, but that another mathematician would define the same mathematical objects, just under different names.

This is not my experience of doing math research. Once a mathematical object has been defined, then we follow a rule system to determine the properties of that object. But I make choices to define a new mathematical object this way instead of that way, and had I made a different choice, I would have a different mathematical object, that objectively (according to our common rule system, at least) can be shown by independent observers to satisfy different properties.

The notion of truth doesn't seem particularly relevant to the distinction between discovery and invention when I think of it this way.
posted by eviemath at 4:10 PM on March 11, 2017 [3 favorites]

Yeah, for my part I'm not interested in some sort of transcendental truth. The question I'm interested in is something more like, once you've said Discovery: there's a physical object that I can go observe. and have consequently at least asserted that there are real physical objects, have you already committed yourself to a form of reality in which some low-level mathematical primitives—positive integers in practice in this case, because flabdablet has taken the position that, among other things, numbers do not exist—can be said to in some sense be real in the same way that "physical" "objects" are?

It has always seemed to me like an important sort of question to tackle before moving on to asking whether objects from tensor calculus or finite-state automata, or other mathematical objects which would appear to be more "complicated"? if that would be a relevant term? can be construed as either discovered or invented.
posted by XMLicious at 1:27 PM on March 12, 2017

flabdablet has taken the position that, among other things, numbers do not exist

Again, this is talking-past.

I have not taken and would never take such a position.

My position on mathematics in general and numbers in particular are that they certainly do exist, in the form of activation patterns in human brains (and in other forms in other kinds of representation engine). They are fundamental to many of the highly effective lossy-compression methods that such engines use to make their representations of reality fit within their own tiny portion of it.

What I specifically deny is that numbers, along with all measurements expressible in numbers (such as mass, location, pressure etc), exist anywhere but internal to the workings of representation engines.

I assume that when we say that elephants are real, we aren't specifying that the word "elephant" exists in English with its grammatical variations in human minds, but that the type of thing which the English word "elephant" refers to in the external world exists, independently of English itself such that there were elephants long before English existed. (And if the world is lucky, elephants will also still exist when English no longer does within any human minds.)

When we say that elephants are real, we're making a statement about reality: that certain parts of it are identifiable using the attributes generally understood to apply to elephants. And those attributes in and of themselves exist, to my way of thinking, in exactly the same way elephants do. They themselves have distinguishability, location (inside the representation apparatus of assorted kinds of representation engine, of which we are one kind), number, duration and (less usefully and rather more difficult to measure, usually) mass and energy.

Attributes of reality are all, to my way of thinking, artifacts of the process of representing it. When we point to some interesting chunk of reality like a group of four elephants, and we say that this group has mass and has number and has locations and has behaviour, that's different - to my way of thinking - from saying that this group is mass or is number or is location or is behaviour.

The way I see it, elephant-like existence is really the only kind that makes sense; to say that attributes (i.e. representation-engine artifacts), let alone abstractions (artifacts of artifacts) exist "in some sense" independently of the representation engines that construct them does more violence to the idea of existing than I have ever found useful, and I have yet to see any kind of unpacking of the "in some sense" handwave that makes any sense at all.

for my part I'm not interested in some sort of transcendental truth

And for mine, I'm not interested in some sort of transcendental existence :-)
posted by flabdablet at 10:23 PM on March 12, 2017

Dude, you literally said

...there is a fairly widely held opinion that numbers, sets, ideal geometric forms and so forth really do have some kind of existence independent of the human discipline and methods we've developed to describe, categorise and investigate them, and could fairly be considered real things even if we'd never actually developed any such discipline.

It's an opinion with which I strongly disagree...

It's not talking-past when I speak of things existing, or not existing, in the same way you yourself initially talked of their "existence". I am confounded as to why I have to keep repeating myself that this is what I mean, when it seems that only you have been doing the juggling with terminology. Yes, when humans think about things, stuff happens in their brains, and that stuff is real, but that has never at any point been what I'm referring to.

By all means, have this internally-existent brain-pattern-only sense of "exists" that you use, but please stop acting as though I have at any point denied that patterns in brains don't exist as patterns in brains.

I mean, how is it not completely obvious that your quoted statement above concerning the existence of numbers is what I was referring to as the position you've taken? Vociferously asserting again and again and again something I've not only never denied but have affirmed whenever you've asked is starting to feel like some sort of rhetorical gambit.

You have just said both that

The way I see it, elephant-like existence is really the only kind that makes sense...

and also

And those attributes in and of themselves exist, to my way of thinking, in exactly the same way elephants do.

How does it make sense to say that elephants and attributes exist in exactly the same way, but to refer to elephants as real: "...elephants exist. There are real elephants." while subsequently parenthetically stating that attributes are "(i.e. representation-engine artifacts)"?

Assuming that the attributes will turn out to not exist in exactly the way elephants do after all, my next point would be that unless two real elephants can occupy the same location, location would appear to be more than something used only "inside the representation apparatus" the way an elephant's name might be, but that again there's a connection to something outside the human mind if even in the absence of any humans, and thereby the absence of any elephants being handled by representation engines along with their external existence, two elephants still cannot occupy the same location.

And besides that, is an elephant's head located between its ears? Or would a thing with the same parts but different relative locations possibly be a different thing than an elephant? (I'm not describing an elephant raising its leg up or swishing its tail, of course, but something more like swapping the left hind leg and the trunk and putting its eyes on the end of the trunk, of something which is alive.) Location would seem to be a bit more than just an attribute of the elephant itself, but appears to be integral and necessary in a variety of ways to what an elephant is in the first place.

No transcendental existence of location is needed, just the normal sort of existence and realness like we've been talking about this whole time...
posted by XMLicious at 12:19 AM on March 13, 2017

I have yet to see any kind of unpacking of the "in some sense" handwave that makes any sense at all.

It means that the way we would represent it when thinking or speaking or writing about it may be distant from the precise way it exists in external reality. If that's a handwave then almost the entirety of what you're describing would also be a handwave.
posted by XMLicious at 12:35 AM on March 13, 2017

this internally-existent brain-pattern-only sense of "exists" that you use

What word would you rather I used when referring to things that exist only as internal brain patterns?

How does it make sense to say that elephants and attributes exist in exactly the same way, but to refer to elephants as real: "...elephants exist. There are real elephants." while subsequently parenthetically stating that attributes are "(i.e. representation-engine artifacts)"?

They're real representation-engine artifacts, really existing inside real representation engines.

please stop acting as though I have at any point denied that patterns in brains don't exist as patterns in brains.

I don't believe I have been acting that way. I don't believe that you have ever issued any such denial.

No transcendental existence of location is needed, just the normal sort of existence and realness like we've been talking about this whole time

Where is location?
posted by flabdablet at 5:21 AM on March 13, 2017

Dude, you literally said

...there is a fairly widely held opinion that numbers, sets, ideal geometric forms and so forth really do have some kind of existence independent of the human discipline and methods we've developed to describe, categorise and investigate them, and could fairly be considered real things even if we'd never actually developed any such discipline.

It's an opinion with which I strongly disagree...

The disagreement is with the assertion that numbers, sets, ideal geometric forms and so forth exist independently of us. I have never said, nor meant to imply, that they don't exist at all.

Nor do I mean that numbers of things and sets of things don't exist. So when you point to four elephants, and assert vigorously that they existed before you pointed to them and will continue to do so when you're done pointing to them, I will naturally agree that this is very likely to be the case. What I won't agree with is that four itself existed before somebody dreamed it up or will continue to exist after the likes of us are gone.

I would expect you to find this position slightly less unpalatable in the case of ideal geometric forms than in that of numbers.
posted by flabdablet at 5:46 AM on March 13, 2017

I don't believe I have been acting that way. I don't believe that you have ever issued any such denial.

In that case then, there seems no point in your repeated assertions that you have not denied that things at least exist as activation patterns in human brains, if I also have not proposed that the things we're talking about do not at least exist as activation patterns in human brains.

I don't even know, at least from the amount of time I've invested trying to figure it out in recent days, how either of us would express the concept that particular things don't exist as activation patterns in human brains, because they'd have to at least exist in our own brains in the process of forming a statement about it. So I am unsurprised that neither of us can point to a case of that claim having been made anywhere in the thread.

And consequently, your repeated assertions concerning the subject feel to me more like some kind of filler, just repeatedly prompting me to reaffirm again and again that it's the case and that you haven't denied it and that I agree, rather than anything which has bearing on the ostensible topic we're discussing.

Of course, if we both agree that thoughts in human heads exist as thoughts in human heads and neither of us has ever denied it, then of course when I specify that something exists in a way that I'm expecting you to disagree with and momentarily drop one in the course of juggling all of the qualifications about internal and external and brain-activation patterns and everything, I'm still talking about existence in the sense you mentioned numbers having in your original statement which you said you strongly disagree with.

Though it's not always a failure-to-qualify on my part: I don't care whether you use the word "exists" to describe thoughts, that's precisely why the sentence fragment you quoted above (at the beginning of this comment) is preceded by "By all means, have" [that sense of the word]. You appear to have taken the sentence to mean exactly the opposite of what it did.

So in conclusion I am still baffled as to how we have ended up spending so much time affirming and re-affirming that we each are speaking of existence in fashions neither of us have deviated from.

What I won't agree with is that four itself existed before somebody dreamed it up or will continue to exist after the likes of us are gone.

Then why do we not have evidence of events having been unconstrained by there ever being four of anything—unconstrained by there ever having been things with the properties which in a contemporary scenario would result in us describing them as "four" instead of "three" or "five"—at times in the past before humans existed, and why is it that events we are not observing never appear to have outcomes that would indicate that there can't be four of something outside of humans watching and handling them in their representation engines?

Where is location?

This seems like a misapplication of concepts—an actual case of an artifact of human perception or thought, because the answer, whatever it is, will not have any effect upon things outside of human minds in the way that location itself not existing externally or numbers not existing externally would, the way the names we give to elephants or planets or the English word "location" itself has no external effect. But if I had to answer, I would say that, since the way we know it exists is that it affects the interaction of what we're calling objects and the interaction of other phenomena, location exists everywhere that location has an effect.

So the same way that we might say a particular instance of another nonmaterial thing, like a particular "force", is located where it causes effects on other real things, we could say that location exists everywhere things can be located.

But as I said, since I think that giving a different answer such as "location is a kind of real thing which affects the external world but which does not itself have location" does not produce any difference which could be verified or disproved in the external world, this question and the answers to it would be things which only internally exist in human minds—a state of affairs which we have repeatedly agreed is possible—as questions and answers about English grammar, for example, would appear to be.

This is actually a pretty good illustration of what I mean by making the qualification "in some sense" when speaking of things existing. Whatever location is, it's the part of it that results in it not being possible for two elephants to be in the same location which is real and externally-existent, not the answers to the questions "What does the word for 'location' sound like in English?" or "Where is location?"

And similarly, science during the past century-plus has revealed to us in a verifiable way that absolute location independent of inertial frame and general relativity effects, rather than relative location like the elephant's head being between its ears, is an artifact of our smaller-than-a-galaxy, significantly-slower-than-the-speed-of-light-frame-rate perception and other limitations on the way we perceive and think about things.
posted by XMLicious at 9:13 PM on March 13, 2017

fwiw, i found it interesting that mathematical understanding might be better, uh, understood with neuroscience: "But what do the bizarre truths revealed by mathematics actually mean? Unlike the truths of physics they can't exactly be touched and seen. Can some of these such as the perceived differences between two kinds of infinities simply be a function of human perception, or do these truths point to an objective reality 'out there'? If they are only a function of human perception, what is it exactly in the structure of the brain that makes such wondrous creations possible? In the twenty-first century when neuroscience promises to reveal more of the brain than was ever possible, the investigation of mathematical understanding could prove to be profoundly significant."

also btw...
This Man Is About to Blow Up Mathematics - "Harvey Friedman is about to bring incompleteness and infinity out of quarantine."

After earning his doctorate in mathematics at the age of 18, he became the world’s youngest professor, according to the Guinness Book of World Records. Friedman went on to teach philosophy and math at various universities, including Stanford University and Ohio State, before retiring in 2012. He now lives in Columbus, Ohio, with his wife of 24 years, Judith Schwartz, a retired psychotherapist. This July, he’ll head to Philadelphia, where researchers at the University of Pennsylvania’s Imagination Institute will scan his brain, along with those of another half a dozen polymaths.

Throughout his extraordinary career, Friedman has never forgotten his first step toward the foundations of mathematics, which proved to be richer (a greater “thrill,” in his words) than he imagined.

Early on, Friedman understood that discovering concrete examples of mathematical incompleteness among already-existing statements would be an arduous task. There’s the continuum hypothesis, the Paris-Harrington theorem, some types of determinacy—but they’re few and far between. So he set out to write his own using a theory he built, called emulation theory. It uses objects from the natural core of mathematics: rational numbers, or fractions of two integers. Rational numbers exist at a very low level of the set-theoretic universe, and mathematicians feel perfectly comfortable with them. But through emulation theory, Friedman revealed a stunning, hidden complexity in them—and a path to the land beyond ZFC...

In this sense, Friedman’s project is as much about the philosophy of math as it is about math itself. If Friedman’s theory unfolds the way he hopes, mathematicians will become entangled with foundational questions, not because of some prior commitment to set theory, but because those questions will emerge naturally in their work. “One goal here,” says Andrew Arana, a philosopher at Pantheon-Sorbonne University in Paris, “is a break between old mathematics, which didn’t regularly encounter results independent of set theory and require large cardinals, and new mathematics in the future, which does.” Higher notions of infinity, and statements about their consistency, would be relevant to mathematicians not otherwise studying infinity—and it could inform their work. Juliet Floyd, a philosopher at Boston University, describes it as bringing philosophy to life. “It makes it something more than just an opinion,” she says.

With a broadened foundational diversity may come new opportunities to solve old problems. In his 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” physicist Eugene Wigner recalls a student asking a perspicacious question: “How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nonetheless, explains just as many phenomena as the present theory.” Wigner goes on to note that the idea is a valid one—or, at least, that there’s never been any evidence to suggest this wouldn’t happen.

The same potential may be present in emulation theory. Although ZFC has more than sufficed, so far, for much of what mathematicians are interested in, that doesn’t mean it’s the best framework at their disposal. The solution of mathematics’ greatest unsolved problems (Goldbach’s conjecture, the Riemann hypothesis, the twin prime conjecture, among others) could require something beyond ZFC—namely, large cardinals and equivalencies to Gödel-esque statements. “Even ordinary statements like the Riemann hypothesis could be equivalent to meta-mathematical statements,” Arana says (although for his part, Friedman does not believe this to be likely).

otoh...
Paradoxes of randomness: "some mathematical facts are true for no reason, they are true by accident, or at random."*
posted by kliuless at 3:45 AM on March 14, 2017 [1 favorite]

why do we not have evidence of events having been unconstrained by there ever being four of anything

why is it that events we are not observing never appear to have outcomes that would indicate that there can't be four of something

Covered here:

Nor do I mean that numbers of things and sets of things don't exist. So when you point to four elephants, and assert vigorously that they existed before you pointed to them and will continue to do so when you're done pointing to them, I will naturally agree that this is very likely to be the case.

And I'll agree that much of the mass of the universe consists of helium which itself consists of nuclei containing four nucleons, and on and on. I have no issue with using the number four, or any other number, to describe and identify parts of reality in order to be able to think sensibly about them and discuss them.

But fourness and location and mass and so forth are descriptions; they are not the thing described. So until some entity evolves that actually makes descriptions/representations of these massive, separable things-in-fours, four itself doesn't exist, and location itself doesn't exist, and mass itself doesn't exist. Only those things that have fourness or location or mass (or thingness!) exist.
posted by flabdablet at 9:26 AM on March 14, 2017 [1 favorite]

Then why do elephants have real, external existence? What is it about elephants that makes them more than an internally-existent-only "thingness" description of the external world, which JiA's field-based lifeform presumably might not use?

And as a separate question, isn't there some salient difference between the description of a particular elephant as "Joe the elephant" versus the description of that same elephant as "an elephant located in Colorado in North America standing on the edge of the Grand Canyon"? Because you can switch from describing "Joe the elephant" to "Batyr the elephant" without any contradiction with reality external to the human mind, but if you describe it as "an elephant located in Kazakhstan in Central Asia at the Almaty Zoo"^{_^} then it can't fall into the Grand Canyon any time soon. (And the process of it falling into the Grand Canyon soon or otherwise would entail significantly different descriptions of changes in location.)

I mean, you can change the names of places, you can change things like whether you're using a radial coordinate system instead of a rectilinear coordinate system for somewhat precise specifications of relative location, you can change the origin of the coordinate system, you can change between feet and meters and between radians and degrees-minutes-seconds, and probably can change lots of other things which are internally-existent-only features of the way that location is represented in the human mind, but if you change the description of location such that in the first case the elephant can fall into the Grand Canyon but in the new case it can't, your location-description comes to be at odds with reality.

Different location-descriptions are tied to reality in ways that different names aren't. Switching from one location-description to another necessitates a change in the location-descriptions of all other things and phenomena the elephant might interact with to maintain location's capability to describe the elephant's interaction with other things and phenomena which have the property of location, in a way that changing names doesn't.

So of course, the thing I'm talking about being real and externally-existent is that thing or phenomenon in the external world that places constraints on the manner in which we can use our internally-existent location descriptors while being "accurate", in correspondence with our examination of the external world, as opposed to the descriptors themselves. Similarly with fourness, there's something in the external world which has an invariable relationship to what we might call threeness and fiveness and all other numbers, however we might internally represent that thing or phenomenon that exists in the external world.

Note that, particularly with the aforementioned varying types of relationships between different location-descriptions and different number-descriptions, locations and numbers actually can, themselves, be described in comparison to each other, and are described in these ways by humans, particularly mathematicians; and also by physicists, who for example describe the varying general relativity properties of location itself with the metric tensor. I'm not quite clear what you were conveying with your sentence about entities evolving that "actually makes descriptions/representations" of some of the things we've been discussing.

So in the case of location, whether you want to call it "location" or "the ability to have location" or something else, it has external existence beyond solely being an internal mental device or artifact of perception in human minds. Hence my qualifications about external existence "in some way" or "in some sense".

Also, beyond names there are immaterial things or phenomena that are expansive or pervasive like location but do not have external existence beyond the human mind, such as P.K.E., "psychokinetic energy" from the movie Ghostbusters, as in Well, let's say this Twinkie represents the normal amount of psychokinetic energy in the New York area. Based on this morning's sample, it would be a Twinkie... thirty-five feet long, weighing approximately six hundred pounds.
posted by XMLicious at 10:32 AM on March 15, 2017

Then why do elephants have real, external existence?

They demonstrably, observably do. Elephants are phenomenal.

Elephantness, though - the abstracted bundle of attributes that we use to recognize, characterize, represent, describe a particular phenomenon as an elephant - does not.

That's the key distinction I'm making here.

beyond names there are immaterial things or phenomena that are expansive or pervasive like location but do not have external existence beyond the human mind, such as P.K.E., "psychokinetic energy"

The fact that human minds can invent fictional phenomena, and can also ascribe attributes with no predictive power to non-fictional phenomena, does not alter the status or location of attributes with predictive power.

I have yet to encounter a need to suppose that any kind of attribute has existence external to minds (or, as I've been describing them more conservatively, representation engines) in order to form a coherent and useful worldview. Which is why, per Occam, I've left that supposition out of mine.
posted by flabdablet at 10:26 PM on March 15, 2017 [1 favorite]

locations and numbers actually can, themselves, be described in comparison to each other, and are described in these ways by humans

Failure to exist outside human brains would be no barrier to this.
posted by flabdablet at 10:28 PM on March 15, 2017

They demonstrably, observably do. Elephants are phenomenal.

We're back to "just because" here. If an elephant trampling you because it's in a nearby location to you, unlike an elephant you can see at a great distance, furnishes no demonstration or observation adequate to offer evidence that location directly corresponds to something in the external world outside your mind, and is not a superfluous internal-only device, there doesn't seem to be any interaction at all you could have with an elephant which would demonstrate its own external existence.

The fact that human minds can invent fictional phenomena, and can also ascribe attributes with no predictive power to non-fictional phenomena, does not alter the status or location of attributes with predictive power.

The point is that in psychokinetic energy level we have something which we know for certain is completely invented by humans because it was created for a work of fiction, and there are obvious differences between it and location or number or mass. Why would you ever have to protest that things which don't exist do not affect other things which don't exist?

Whatever semantics you want to use in human language to describe the latter things above, and however you want to position or bundle them in your mental object model: whether ephemeral, temporarily-existent elephants are what you want to pose as things which location and mass and number are attributes of, or whether an elephant is just a configuration that mass and energy is temporarily located in and which other more-permanent things adopt momentarily; the simple fact of the matter is that two elephants could not occupy the same location and three elephants could not fall into four pits simultaneously even if humans never existed or never made predictions or never even thought about it.

Unlike an elephant's name or PKE level, no human invention is required for the external existence of those things to be manifest, and humans or even aliens who had never met one another would not need to confer beforehand or exchange inventions in any way to independently agree that two elephants can't occupy the same location and that three elephants can't fall into four pits simultaneously. It just doesn't have anything to do, whatsoever, with the way you want to represent it in your human mind or whether it's useful to you or not.

Though, it's kind of bizarre that you seem to be saying that the mental representation of an elephant with just a name and PKE level is no more or less useful than, that there's no more "need to suppose" the mental representation of an elephant with location, mass, and all of the other externally-existent attributes or phenomena or whatever humans might call them.

It's not some minimal-assumption application of Occam's razor to say that these things have no existence outside the human mind if you can't demonstrate external reality behaving as though it can do without them, the way that it can do without PKE or names. It's merely an inadequate explanation the way that "It's magic!" would be, even though that's a still-simpler explanation with even fewer assumptions.
posted by XMLicious at 6:38 AM on March 16, 2017

If an elephant trampling you because it's in a nearby location to you, unlike an elephant you can see at a great distance, furnishes no demonstration or observation adequate to offer evidence that location directly corresponds to something in the external world outside your mind, and is not a superfluous internal-only device, there doesn't seem to be any interaction at all you could have with an elephant which would demonstrate its own external existence.

But it wouldn't be trampling me due to being in a nearby location; it would be trampling me due to being nearby. All I need in order to evaluate the likelihood of being trampled is a good estimate of my distance from the elephant. Trampling works equally well in the Melbourne Zoo as on the savanna.

Location is a far more complicated idea than distance, which is itself an abstraction. A thing's location can only ever be defined on the basis of its distances from some set of arbitrarily chosen reference points along some arbitrary set of paths. Anything I observe can have at least as many locations as I can invent reference points to base them on. Einstein's little rigid rods and clocks and reference-mollusc are strictly imaginary.

The fact that two elephants can't occupy the same location is a deduction from the categorization of elephants as large physical objects with boundaries sharply defined relative to their size, and the rule that large physical objects with sharply defined boundaries will not, in general, permit the distance between those boundaries to become less than zero.

The notion of location is an abstraction we use to generalize that deduction. It's not necessarily a well-defined attribute of any physical system. Saying that location in and of itself exists external to the mind, as elephants do, is just a category error.

Elephants don't fail to overlap because they can't occupy the same location; that's exactly backwards. They demonstrably fail to overlap, and that primary observation is the basis for the invention of the useful abstraction of location.

the simple fact of the matter is that two elephants could not occupy the same location and three elephants could not fall into four pits simultaneously even if humans never existed or never made predictions or never even thought about it.

I've never given even a hint of disputing that, so I'm a little confused as to why you keep on bringing it up.

humans or even aliens who had never met one another would not need to confer beforehand or exchange inventions in any way to independently agree that two elephants can't occupy the same location and that three elephants can't fall into four pits simultaneously.

I would certainly expect multiple parties acting completely independently to be able to discover that elephants don't, as a rule, overlap and that filling pits with elephants requires, as a rule, as many elephants as pits. Even so, conferring and agreeing on the meanings of the words used absolutely would be necessary in order to reach agreement; otherwise the parties would just spend all their time talking past each other.

it's kind of bizarre that you seem to be saying that the mental representation of an elephant with just a name and PKE level is no more or less useful than, that there's no more "need to suppose" the mental representation of an elephant with location, mass, and all of the other externally-existent attributes or phenomena or whatever humans might call them.

That is indeed a bizarre interpretation of what I've been saying.
posted by flabdablet at 7:36 AM on March 16, 2017 [1 favorite]

I've never given even a hint of disputing that, so I'm a little confused as to why you keep on bringing it up.

Precisely because I would expect that you have no way of disputing it, even though you keep insisting that the phenomena described by those things exist only internally within human minds. If location and numbers only existed within human minds, I would be unable to demonstrate location and numbers having an effect within the external world.

But I easily can make such a demonstration; I'd hoped you would come up with some way of explaining the discrepancy.

I would certainly expect multiple parties acting completely independently to be able to discover that elephants don't, as a rule, overlap and that filling pits with elephants requires, as a rule, as many elephants as pits.

If you concede this then it's pretty much the end of the discussion of my particular question, and demonstrates, to me at least, that these things aren't invented but are discovered in the sense I've taken the entire thread to be about.

This is what I would mean by making the above statements myself: unlike a name or something like PKE levels which identical conclusions couldn't be reached about without previous communication, these are things which exist, in some sense at least, objectively and external to any mental representation of them, an existence independent of the human discipline and methods we've developed to describe, categorise and investigate them to quote your original statement, which completely independent parties could reach the same conclusions about regardless of differences in their disciplines and methods or even whether they're humans or aliens.

And yes, for any thinking beings involved in initially-independent deliberations to subsequently compare their conclusions, some kind of commonality of communication would have to be established later on.

So apart from that, certainly say whatever you would like to about shades of meaning of the English phrases "to exist" and "to be real" and whether "due to being in a nearby location" or "due to being nearby" is proper and what the backwards way of describing things is versus the forwards way and category errors and which things would be attributes and which would be things-attributes-apply-to in particular mental models, and I'm at your disposal to discuss those details if you've got your own questions about them or inquiries into them or would like to discuss anything else.
posted by XMLicious at 9:14 AM on March 16, 2017

even though you keep insisting that the phenomena described by those things exist only internally within human minds.

No I don't. The phenomena described by those things are quite uncontroversially real, as I have repeatedly reaffirmed. Those things themselves exist only internally within minds (or, more generally, representation engines).

If location and numbers only existed within human minds, I would be unable to demonstrate location and numbers having an effect within the external world.

I don't see how that follows.

these are things which exist, in some sense at least, objectively and external to any mental representation of them

I'd find your position more convincing if your could specify their manner of "objective and external" existence without the "in some sense" handwave. Because as far as I can tell, things that only manage to exist "in some sense" simply fail to exhibit objective, external existence altogether.
posted by flabdablet at 9:29 AM on March 16, 2017

Okay... you don't see how it follows that something which only exists in the human mind cannot have an effect on the external world? If that doesn't make sense to you, I don't understand what you think we've been discussing this whole time.

Do you at least see that something which only exists in the human mind can't have affected events which occurred before humans existed? Or why, at least, most people might think this to be the case, like people you're having conversations with?

I still do not understand your misgivings about the expression "in some sense". It means exactly what you're saying here:

The phenomena described by those things are quite uncontroversially real, as I have repeatedly reaffirmed. Those things themselves exist only internally within minds (or, more generally, representation engines).

I have at all times been discussing the real phenomena described by the concepts or brain patterns or representation engine elements or whatever you want to call them, rather than the concepts or brain patterns or representation engine elements themselves. I don't know how I could have possibly made that any clearer.

I do not recall you saying at any point that the phenomena described by numbers are uncontroversially real. I'm not going to dredge through the thread looking, but go ahead and point out an example or examples of that if you have them. I think you probably should have more vocally acknowledged the point where you went from strongly disagreeing that

numbers[...] really do have some kind of existence independent of the human discipline and methods we've developed to describe, categorise and investigate them, and could fairly be considered real things even if we'd never actually developed any such discipline

to believing that they represent phenomena which are "quite uncontroversially real".

Note that my very first response to that statement was to ask what you meant by it, and if you really meant at that point that numbers are just how we think of uncontroversially-real phenomena we didn't invent (to be honest, I'm a bit skeptical that it's true this is what you meant at the time), it would have kept things much shorter, especially in a thread explicitly on the topic of whether mathematics is invented or discovered, to simply say so rather than talking about entities who value predicting things and "No such entities? No numbers either."

But hey, thanks for explaining that the activation patterns in human brains don't exist if human brains don't exist.

Like I said up above, this is pretty much how I expected a conversation about whether numbers are invented or discovered to go, with a sort of M. Night Shyamalan reveal of "we actually agreed all along!" at some point, but I sincerely thank you anyways for going through the paces with me.
posted by XMLicious at 10:42 AM on March 16, 2017

I can't express my position any more clearly than I have. I don't know whether your persistent refutation of positions I've never expressed and don't hold reveals a lack of explanatory skill on my part, a reading comprehension fail on yours, or just two worldviews too far apart to reconcile.

Sorry about missing out on that reveal. But perhaps that's the twist.
posted by flabdablet at 11:50 AM on March 16, 2017 [1 favorite]