What’s The Math Made of, Ding-Dong?

Tegmark etc. might be interesting.

Also: don’t worry about benzene. It has been around a long, long time. Certainly since the dinosaurs started fermenting into petrochemicals. And even before that. It’s a has-been for eons, even if it hasn’t been seen.

Pedagogically, I can think of at least one tenured mathematician who’s done at least some original research of my acquaintance who didn’t grow up as a math prodigy. I mean, odds are your girls aren’t going to be mathematicians, most people aren’t. But I wouldn’t rule anything out.

What LizardBreath said.

Also: if one of the facets of something being “real” is that it can be a cause with effects, then numbers are surely real, as are other less constrained forms of language. I used to be a reductionist — that the universe is matter + energy — but there are clear causal effects of language (including mathematical language) that can’t be reduced to brain circuits. The movements that lead to the typing out of comments in this thread, for example.

I’m sure some of the philosophers here could put this better and tell me what I’m talking about.

So most people who know me will agree that I’m an idiot, but it seems pretty clear to me that there are no numbers past the number one. You can obviously have different multiples of one, and name each one I dunno something like two or three, but that’s basically all trickery.

Five petals? Pshaw, nonsense. It’s 1 or nothing, which I guess means we live in a digital universe. And anyway, you have to exist at our scale to see five petals on the flower instead of a whole mist of loosely-connected molecules that you could drive a bus through, all of them made up of more 1s

And that’s the most coherent my argument gets, I’m afraid. I have literally no math or philosophy chops beyond the pshaw.

A different but related question: this has been bugging me for a long time (years), at last here’s a thread I can ask someone.

When it comes to 3d geometric shapes, scale matters, right? The relationship of volume to surface area changes with scale? So (say) a cube that’s 1cm on each side has proportionally a greater surface area than (say) a cube 1m on each side?

So a platonic ideal of (say) a cube cannot exist, because without actual physical dimensions it’s not really a cube? Because it doesn’t have a relationship of volume to surface area? Unless you express this relationship in an equation, but isn’t that cheating?

Like I said, I’m really quite stupid, so be gentle with me.

What does Platonism about math mean? Is my idea of Platonism screwed up because one of the first philosophy courses I took was history of philosophy, and before the post-epistemological turn, meaning I’m hopeless? I’ll happily say there’s a fact of the matter about 2+2=5 and 1001=1000+1 and right triangles, but I’m dubious about 2 existing in the Mind of God.

Since I’m already offending Belle, I’ll finally work up my curiosity to ask if she went to Columbia or Barnard (since we were there about the same time but IIRC they have separate departments in both philosophy and classics).

I always thought this Hamming talk was better than the Wigner piece.

Maybe because I aesthetically like Hamming’s conclusions better than Wigner’s.

Do complex numbers “exist” the way counting numbers do?

This science radio show I listened to interviewed child development experts and anthropologists. In it, they said that babies (and people in societies which do not use numbers) naturally think logarithmically. And the only reason we use base-10 integers is that our parents count to us as children and we eventually just accept to this different, socially acceptable math. For me, it was *mind blown*.

Listen yourself:
http://www.radiolab.org/story/91698-innate-numbers/

Just when I get comfortable in thinking of math as a purely mental construct, they pull me back in. More seriously, Meillassoux is interesting on this sort of thing, particularly his conceptions of ancestrality, the arche fossil, correlationism, etc.

15: From the Wiki article on Meillassoux

For these reasons, Meillassoux rejects Kant’s so-called Copernican Revolution in philosophy. Since Kant makes the world dependent on the conditions by which humans observe it, Meillassoux accuses Kant of a “Ptolemaic Counter-Revolution.”

This is the leap that makes metaphysics.

Kant does no such thing. He makes the world we can speak of or talk about dependent on categories etc. He says nothing about the “world” because nothing can be said.

There’s a series of Eureka! objects or constructs I dunno what, where the pomposity serious guys are all “We have found the building blocks of everything!”
Then pause, then oh.
So gluons and muons and so ons.
Quark! Then…

Now there’s Penrosian quanta making the brain happen. Microtubules something something.
And a mystery bit coming in from the sun that we didn’t know was there, but it was, all along.
It was molecules for a while way back when. Greek atomic theories of the BCE.

From miasma to bacteria to viruses to prions to “Nothing more to see here, just little bricks and lego-like-things.”
From which the illusion of individual consciousness is built.
Or not. quite. there. yet.
The micro-boundaries of our living being keep getting pushed a little further in and down.

The awesome to me but evidently awful to many idea that it just keeps going.
With fun portals to other dimensions becoming available, possibly soon.

Math guys of a certain stripe evidently hate them infinites, understandably. Lots of doctorish professionals don’t like metaphysics for pretty much the same reason.

Germ theory or some more recent analog helps a lot for genetic engineering and all that, but it’s a mechanical not a wholistic understanding, and it is not the final complete and total what-is-it.

Math is like the skeleton of the thing, it is not the thing.
Skeletons are very important, essential to what we are, but they are not what we are.

Personally, I’m considering the question of the independent existence of integers to be an open one until we meet E.T.s. If the kids from proxima centauri have something recognizable as the Peano Axioms, then yeah, there’s probably something real casting a shadow on the wall that looks like “1, 2, 3…” If not, oops, we were actually looking at a mirror and telling ourselves it was a window. Again.

(Of all of my closely held and cherished crank-like beliefs, this is by far the most cherished and crankiest.)

Like, was there an even or an odd number of hairs on Zoë’s head on March 23, 2006? There’s some true fact of the matter, but it’s indeterminable.

Pretty sure John is wrong here. Even if we narrow the time to an instant, we’ll still have to come up with criteria for whether given hairs are on a given head. (And are we counting peach fuzz? Eyelashes?)

Seriously, as you note with the hexagons of the honeycomb, reality is messy. Even the carbon atoms of the benzene ring form a hexagon only in a sense that needs some cautious defining.

@Martin Bento, #19

It’s called “Adventure Time,” on Cartoon Network, and yes there is more. Probably some free full episodes available here:

http://www.cartoonnetwork.com/tv_shows/adventuretime/video/index.html

#24- None of us exactly see why the macro past is unique but the macro future isn’t. The basic form of quantum field theory is time reversible (ok, CPT, to be precise). Nevertheless, in all our experience that irreversibility is just a fact. We can tie it in with other facts, in particular the second law of thermodynamics, but at best that just makes one big mystery out of a bunch of little ones.
Maybe there’s some anthropic argument, along the lines that Boltzmann suggested long ago. Out of some swirling multitude of existence, only the strongly T-asymmetric parts support certain types of information processing (us). But that’s not exactly settled science.

This actually reminds me of the Holbo post from, last summer?, on utilitarianism where the thread turned into one long back and forth on moral objectivity (well, mostly one long forth against). I’m sort of curious on what the overlap is (if any) of people who are diehard subjectivists in ethics but Platonists about math. Because that is weird, no?

(For what it’s worth, I’m a quasi-Platonist about both, but only because I’m sort of an idealist about everything. No, really.)

I’m usually not a fan of the cop-out declaration that an old philosophical question is a “pseudo-problem,” but I can’t for the life of me convince myself that the question “are numbers real?” has a clear meaning or whether the commitment either way leads to interesting further commitments or consequences.

“Like what if there were a sakura with its five petals, and it were pink, but no humans existed. Would it still be pink? Would it still have five petals?”

Would it be different or less effective an argument to say this just as well demonstrates that words are real, since “sakura” and “pink” and “petals” exist even if there are no humans? It strikes me as equally strange an argument. Aren’t numbers just a more precise and abstract form of language or signs? What does it mean to ask if symbols, signs, or language are “real”?

At best it demonstrates that number-able or distinguishable things exist. And distinguishability doesn’t demonstrate anything about number as a “reality”: I distinguish a face in a newspaper photograph, but the photograph has no “real” number–it’s many dots.

Does being a realist about number commit one to realism generally or being an anti-realist about numbers commit one to anti-realism generally? What are the stakes of such a debate? What do we lose if numbers aren’t real? Not even flower petals, I’d say.

The “are numbers real” question is like a stand-in for “are mathematical principles, formulas and artifacts things which exist in the real world, or are they human constructs” question, which is important since math is the language of science / all kinds of ish is predicted mathematically before confirmed empirically => if math isn’t “real” then what can we say is actually “real”?

The OP brings this all up, incidentally, pretty seamlessly going from “oh hai flower petals” to the “unreasonable effectiveness of Zoë” point (yes I copy-pasted the name because I don’t know how to do umlauts) to BENZENE RINGS WAKE UP SHEEPLE.

When I do maths I do it as an idealist – as if I was looking to discover something that exists independently of us. But my firm belief is that maths are socially constructed just like all human intellectual endeavor. Now this is not to say that in a different society, two flowers with 5 petals would make 11 petals total!

But maths is more than this: it is the practice of studying structure, counting, measurement – and this practice is historically and socially developed. For example for a long time in the mathematical practice that grew out of that of the Hellenistic world, “numbers” as in “five flower petals” where considered distinct from “numbers” as in “the length of this side of a triangle is five” – there was no concept of ‘number’ as we use now that can be used to either count or measure. This concept is a human creation, even if the flower petals are obviously not.

Similarly, the maths we practice as research mathematicians today – the centrality of the notion of mathematical proof in particular – again is very specifically historically developed out of the Hellenistic – Arabic mathematics, and in its current form is very modern; its spread around the world was a consequence of Western colonialism, missionary work (Jesuits brought Euclid’s Elements to China after a Ming Emperor appointed a Jesuit monk chief of the Imperial Office of Astronomy) and imperialism. Traditional Chinese mathematics for example considered proofs an affront to concision and unnecessary luxury; in Japan, Western-style math was introduced all at once during the Meiji restoration as part of the program of Westernization.

@Anon, #29: At best it demonstrates that number-able or distinguishable things exist.

Also innumerable things, like real numbers. Unless there’s a way to say that the natural numbers are different from the real numbers. I haven’t thought about this, aside from reading Quine long ago, but now I’m wondering whether numbers being “real” implies that the universe holds an uncountably infinite number of things and whether there’s any reason to say that numbers came into existence only when the universe did. (Not wondering very hard, though.)

And anyway, one can be a mathematician and discover new things about math without being an “incredible math genius.” Most people are just muddling through their career making incremental progress, and there’s nothing wrong with that, is there? A decade-ish into my academic career I’m pretty much resigned to never doing anything memorable, but it’s a living.

@ 15

I thought you’d never ask!

@ 29 – Aren’t numbers just a more precise and abstract form of language or signs?

That’s what I’m thinking. Symbols are perplexing because they are pointers, not things. The same thing that makes language useful also makes it extremely deceptive. For example, conveying a false sense of permanence and identity. But also shifting our attention from the world of experience to that gray zone of symbol manipulation we call “thought.”

Well, in one sense logarithmic scales are built into our sensory apparatus: we experience both sound and light via log scales. Ask someone “which light is N times as bright?” and they’ll point to a light that is putting out something-to-the-N times as much light. A light putting out four times as much light will appear, meh, maybe twice as bright. That’s also why decibels are log-scaled. (Differently put, our phenomenology is doing log compression).

But it’s a few more steps from that, to saying that we are comfortable doing conscious thinking or conscious processing via logs. That takes a bit more practice.

Whatever you think about numbers, though, you’ll have to grant that there have always been eternal, mind-independent blog-topics, which would have existed even if no internet had ever noticed them.

And Belle, I’m totally with the people saying you should quit telling your kids they can’t do math.

The fact that they have not already shown prodigious abilities is, I think, pretty good evidence that they will not wind up wandering the halls of Fine Tower scribbling gibberish on blackboards, or searching through the dumpsters of Telegraph Avenue.

But it says very little about whether they could have productive careers in the field of math (the very large and very disparate fields of math).

They have as good a shot at making money doing math as most people do, and it is the blameworthy ignorance to think you know otherwise.

Not sure about the sky, but the sheep were in Homer.

My Honours thesis was on algebraic topology. I certainly felt as if that was made up by humans

@ oldster:

The curious thing, for me, is that we would think that we’re translating logarithmic-appearing shifts in our observations – into something like once, twice, thrice. But that begs the question: How do we know we aren’t actually translating base ten language (beliefs? rubrics? schemata?) into logarithms, for the more fundamental circuitry of mind to work with?

Kids (and fans of logarithms) giving the answer that 3 is halfway between 1 and 9 seems to be one data point towards logarithms being related to a more basic mode of thought. Sure, in turn there may be base 10 or differentiable manifolds or whatever underneath that, but it looks like as far as we’ve gone that logarithmic thinking is the more basic, so the skepticism here seems kind of shaky to me.

But the logarithmic stuff is news to me – pretty interesting stuff.

Do blue numbers exist? How about red numbers?

If we posit blue numbers and red numbers, then numbers can have properties that are interesting or boring, depending on your reaction. (Blue and red numbers are in “The Education of TCMITS”.)

On a sphere, a great circle is like a line in a plane, except that if you number the points on a great circle, you can’t say that one number is greater than or less than another, since you can start from one point and reach the other by going in either direction. You can get to positive numbers by starting from a negative number and going towards and then through negative infinity, and vice versa. To get greater than and less than relations, you only have to eliminate from the great circle the point at infinity. That gives relations like ordinary numbers. But you could also eliminate the point at zero. In that case positive numbers are less than negative numbers. Do such numbers exist or not?

Um… Is nobody going to comment on how PB is becoming more and more of a personified allegory for the American police state/empire?

This detail must have somehow slipped my memory when I had watched the episode for the first time, but here she’s outright torturing the Ice King to divulge secret information held by members of a different sovereign power structure… Taken together with the development in the most recent episodes, the show’s really becoming a pretty blatant satire.

Philomena Cunk!

I think that the point is that there is a difference between ‘real’ and ‘true.
There is a reality, that is real.
We create some model to describe this reality.
Models that correctly predict this reality are true.
But these true models are not the same thing of the reality that they describe, and are necessariously very partial.
So relatonships among numbers are true, but are not real, as they represent just a very small slice of the reality they describe, a slice that we selected because subjectively more relevant or easier to model.

Also, do ‘action movies’ exist?
There are some movies, and given a certain definition of action movies, certainly some movies would fall in this cathegory. Yet the cathegory in itself is arbitrary.

Ditto science fiction. In retrospect we call The Time Machine science fiction but that term didn’t come into existence till much later.

‘Film noir’ didn’t exist until French critics invented it long after the major works had been produced.

And Lord of the Rings was a ‘fable’ like Animal Farm until ‘fantasy ‘ established itself as a marketing category.

Lee Arnold @57 (currently): I think we strongly agree here. There are probably deep structures to the universe (even if such “structures” are underneath merely statistical approximations of the range of outcomes of a bunch of ultimately chaotic behavior blah blah blah), and it’s possible that “arithmetic” is only one of a possible number of (potentially mutually contradictory or at least not obviously overlapping) means of explaining those structures.

But it’s worth noting that this is very much not the position taken by the “strong mathematical platonists”, such as Belle above. (Steven Landsburg also comes to mind as such a proponent.) The position there is that, for example, the natural numbers exist in the same way that a rock exists, because integers describe a property of the universe so fundamental that the rock could not exist without them, and any intelligent species capable of in some way describing a rock would of necessity derive the existence of the natural numbers along the way. To put it mildly, I find this undersupported by the available evidence.

neat post. I suck at Math, and this was very educational. The sad thing is that I am pretty sure Math still exists outside my cognition, but I am basically mathematically illiterate (innumerate?) so I would have no way to prove its existence or non-existence (cant even manage much algebra). It would be like arguing about the Bible, or Plato’s Republic, if you couldn’t read. I suppose there might be a way for me to learn Math, but I’m in my forties and kind of pressed for time as it is.

Do you say you are a Platonist about numbers because you believe in numbers as mathematical objects? Or do you go whole hog and affirm the existence of the eidetic number, which mathematicians know nothing of?

Badiou wrote a book on this -“Number and numbers”. Unlike even “Being and Event” which I found hard, but doable in small chunks, this one I gave up on about 100 pages in, as I found my brain becoming number and number.

The cartoon reminded me of Clarke’s “Any sufficiently advanced technology is indistinguishable from magic.” When my “Science Fiction and Fantasy” class tries to determine the differences between the two genres I bring up that one. “What if Gandalf just uses nanotech?”

Belle: if we’re ranging into the realm of “my crankish ideas projected onto the canvas of dodgy sci-fi writing”, I’ll aver that “only using irrational numbers” or whatever strikes me as a kinda cheap dodge. I have a sneaking suspicion that if we ever met spacefaring ETs, we might not recognize anything they do as “math” or even “cognition” at all. The only writer I can think of recently who tried to take the idea seriously in a “hard” SF context is Peter Watts in “Blindsight”, although Watts was more concerned with the question of consciousness rather than math qua math.

(You can download Blindsight for free from Watts’ website — I’m not linking only to avoid getting eaten by CT’s increasingly erratic anti-spambot. Google will lead you there, and you totally should.)

“I called you naughty boy because I do not like that other world.” …Martha’s letter to Henry Virag, Lotuseaters

“I do not like that other world she wrote. No more do I. Plenty to see and hear and feel yet. Feel live warm beings near you.” …Bloom, Hades

Das Gefühl der Welt als begrenztes Ganzes ist das mystische.

Es gibt allerdings Unaussprechliches. Dies zeigt sich, es ist das Mystische.

Wovon man nicht sprechen kann, darüber muss man schweigen.

One might start with the question of whether mathematics, beyond the most elementary operations of finite counting and perhaps simple arithmetic, would be possible without symbols, those peculiar combinations of both signifieds and signifiers, which are not just a material convenience or crutch, but are required to designate the objects of thought in the first place.

Thing about this tho is that it’ll commit you to anti-realism about non-observational (“theoretical”) entities in science—like, I don’t know, quarks or whatever. Which obviously is a position held by many people in the area, but always seemed to me a bit drastic.

More generally, I think that at least part of what motivates Platonism about math is a recognition of the necessity of mathematical truths—their obvious non-arbitrariness. (Which of course isn’t quite the sort of motivation BW is presenting in the post—at least I think not?) It would be I think a total disaster if ‘humans invented math’ were taken in a sense whereby it implied that mathematical truths could be, e.g., changed at will. (We made it; we can make it otherwise! That obviously seems a nonstarter.) All that said, I find that there are better ways to account for the non-arbitrariness of math—its, umm, a priority—in ways other than a bald Platonism. Tho it’s not much help for the metaphysically weak-kneed.

Fucking math, how does it work?

“2 + 2 = 5” “square circle”

(presuming mathematics is a language)

1) It is a marvel of language that we can create expressions or statements that “represent” the impossible, unimaginable, or the unthinkable that are such because of the intrinsic or internal nature of language. These kinds of statements are not of the “there is a unicorn” kind.

2) Lord I don’t know any Cantor. But this means to me that perhaps language is of a cardinality greater than either perceived phenomena or human thought processes, and so language cannot be mapped onto, completely and reliably, and is incommensurable with either.

3) I also feel that we are falling into some weird variation of Anselm’s ontological argument here, with 1 & 2 above being a parallel attempt at a historical refutation.

Anselm defined God [mathematics] as “…that than which nothing greater [different] can be conceived,” and then argued that this being could exist in the mind.

So I’m a mathematician and definitely not a Platonist, largely through experience and the fact that, on reflection, Platonism just doesn’t really make any sense to me.

For instance, I honestly don’t have an opinion on whether the Axiom of Choice is true or not. Its convenient, and thats about all I am willing to commit to. But the question of whether it is true seems somewhat incoherent.

Let me argue by analogy. Ask yourself whether “en passant” is a valid move in chess. Yes, of course it is. Now ask whether you want to include it if you re-design chess. Well, you might have an opinion on that, and thats cool. Now ask yourself, in the middle of a re-design, whether it is “true” that a pawn can capture as in “en passant” and whether your rules should therefore reflect that. Thats more or less what mathematical questions about truth look like to me – “truth” looks pretty nonsensical here. Your opinion about the validity of the rule might be so strong that you desire to claim it as “true”, but it is hard to see this as more than a rhetorical flourish. You are just pounding the table really hard while trying to convince me.

As to Belle’s question….if I interpret “+” as “add the next number plus one” then “2+2=5”. Easy, done. Conceivable. I just conceived it! And I am certain that that is absolutely not satisfying. Why? Because you want to keep the natural bits of arithmetic, whereas I just arbitrarily introduced a change (which arguably isn’t good for aesthetic reasons or whatever). But ok, but then if I keep the natural bits and the way they fit together, then saying that “2+2=4” is just an expression of those rules and the way they fit together. It is true, in the same way the pawns may take using en passant by standard rules of chess is true. And, of course, we can imagine different ways to construct the rules of chess, just as for mathematics (although most people don’t quite realise that, but Godel is knocking around here, challenging your notions of “truth”). It is also the case that people have an intuition about arithmetic that makes it feel as if the objects are real….but I think that says far more about human psychology than anything else.

No, you’re both very intelligent children, you can learn calculus just as well as anyone, but if you were going to be an incredible math genius or something we’d kind of already know. Sorry.

Half of my Facebook friends are active mathematicians (that is, if I were on Facebook) and a number are really top class (what would be a credible measure to an outsider, oh I know, they have a Wikipedia page!). Among them, two or three were child prodigy. The rest is average chump all the way.

but sometimes mathematicians come up with stuff that’s crazy. Like string theory

Craziness credit should be given where it’s due: physicists came up with string theory. Surprisingly, they have used it to prove a bunch of interesting theorems in math but have (for the moment) mostly failed to do physics with it.

[C]an string theorists not propose any test whatsoever that would prove their hypotheses or is it rather the case that we lack the capacity to perform the tests that would figure it out?

I don’t think there is a scientific consensus on the answer to this question. But mathematicians doing mirror symmetry don’t care either way.

if we stipulate that humans invented the concepts 2, addition, equality, and 5, could they refine or alter their ‘invention’ in such a way that 2+2 did, in fact, equal 5?’ Obviously, totally, self-evidently not, right?

2+2=5, no. But 2+2=0, why not? That’s how computer count after all. Anecdotally, as the other thread was utterly ruined by my Barbie quote, I was doing math in an MRI in order to test whether math intuition was based on the neuronal recycling of (some parts of) the visual neocortex. That’s Stanislas Dehaene position. Or maybe the recycling is mediated by the language areas. That’s what Elisabeth Spelke thinks. In both cases, one can imagine fundamentally different cognitive properties (say, those of a bat, to take the usual example) leading to a radically different mathematical intuition than the human one. What’s math made of, ding-dong? How about the neuronal recycling of our ability to recognize faces?

Thought experiment:

Is it possible to conceive ratios without numbers?

Well, she’s gone now, but when Belle says

I believe numbers and mathematical operations can be said to exist in some meaningful sense independent of human intelligence

that’s fine, but if you don’t say exactly what sense that is – which is *very* difficult – then you can unwittingly sow the seeds of some nasty and persistent confusion later on, of which the (cap-P Platonic) Great Chain of Being is only the most extreme (though interestingly not the worst) example. That’s all. Probably won’t come up, but you never know.

Since Zoe’s learning Japanese, I wonder, whether Japanese has the thing Korean does [1] where you don’t say a number, you have to add a syllable to say what kind of thing it is after the number (people, slips of paper, whatever)? Korean also has the issue w/r/t number words that there are two sets of number words and they’re used for different things even with the same syllable suffix, which Japanese most likely does not. It occurred to me to wonder whether that sets up different issues about abstracting numbers from concrete sets.

[1] In Korean I can say hello and thank you and “two tickets” and that’s about it.

I wonder, whether Japanese has the thing Korean does

It does, even the thing you thought would most likely not happen.

It occurred to me to wonder whether that sets up different issues about abstracting numbers from concrete sets.

I don’t think it does, though, because that linguistic phenomenon is probably a syntactic reflex of the fact that numeral “adjectives” are not usual adjective at all, and that syntactic reflex is present in many languages, including English and Romance (think about rigid placement, overlap with determiners and absence of inflection in languages where adjectives are normally inflected, say tense inflection in Japanese or number and gender inflection in Romance).

It occurs to me that I’ve only heard about platonism because of physicists and I’ve only heard of constructive mathematics because of computer scientists.

It’s not actually surprising that mathematicians themselves don’t particularly care about foundations–they’re content to play in their system of axioms. I don’t actually want to think about whether my appreciation of a work of art is based on any sort of objective value while I’m actually in the museum. But once you actually want to do something with math, some stuff becomes nice to know. If you define a theory of planetary motion using mathematics, do you have to worry about mathematics working differently on a different planet? If you want to implement mathematical theory in a computer proof assistant (which you would like to do because all of your physicist friends tell you that mathematics is required to understand the physical world and you want to make machines that solve real problems) then arguing over foundation is literally arguing over choice of programming language–and choice of programming language actually matters a great deal in an engineering project.

These are not non-problems. I only maintain a recreational interest in this stuff, but I think there’s some exciting stuff happening in constructive mathematics. A practical use for platonism is much harder to find, of course. But as I (naively) understand it, physicists are interested in string theory more because of it’s mathematical beauty than any falsifiable predictions made thus far. If string theory could later make such predictions, and the predictions held true, would that not be evidence for some kind of small-p platonism? If it is not merely the consistency, but the beauty of mathematics that makes it epistemologically useful, that suggests that the kind of structure we use mathematics to describe is somehow prior to physics.

Thought experiment: Is it possible to conceive ratios without numbers?

For sure. Cribbed, it’s said, from the earlier work of your man Eudoxus, who was sort of Carl Perkins to Euclid’s Elvis.

Eudoxus : Euclid :: Carl Perkins : Elvis

I see what you did there!

Instead of asking if numbers exist we should define existence in a way that is mutually agreeable and then define numbers in a way that is mutually agreeable and then ask if the the former is a condition of intelligibility for the latter. Or something.

(Perhaps one could make up a number-suffix that meant `dimensionless’?)

For example for a long time in the mathematical practice that grew out of that of the Hellenistic world, “numbers” as in “five flower petals” where considered distinct from “numbers” as in “the length of this side of a triangle is five” – there was no concept of ‘number’ as we use now that can be used to either count or measure.

This perplexes me, as I believe I could (in spring) measure by laying flower-petals along my line and counting them. Yesno?

Also, it doesn’t actually show that ‘2 + 2 = 5’ is conceivable. Any more than I can make true that the Moon is made of green cheese by decreeing that I’ll use ‘the Moon’ to refer fried balls of green cheese, let us say.

I’ve never understood how Max Tegmark can propose the existence of universes of platonic entities with no evidence whatsoever, and have his work published in Scientific American, while Rupert Sheldrake’s theory of morphic resonance, which at least has some evidence for it, is ridiculed by the scientific establishment.

@js, 112:

I think I haven’t explained myself well. I’m not claiming that the mind-dependence of the operations in question is settled by my silly example, but I am trying to point out that the question of truth is kinda moot, since the answer is already baked into the question. That is, my silly answer is totally unsatisfactory because you already have a strong sense of what the symbols must mean. And that meaning pretty much includes taking 2+2=4 as part of the package. You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them.

A technical question: does mathematics have a semantics, or just a syntax? (I don’t know the answer, but am just wondering what those who do mathematical logic or meta-math would say).

At any rate, Belle’s worry seems to me to be the sort of thing that Rorty would respond to with, “if it don’t itch, don’t scratch”. I could be that one could make sense of “2+2=5”, via ambiguous re-definition. (It was a more sophisticated version of such that got this round of the business going). Adding logs is the same as multiplying integers, so some similar expression could occur, if one could somehow “count” with logs or define what the “unit” would be. But the worry is the same for language. Perhaps someone might be tempted to speak of “dry rain”. (What would that be? Sleet?) But the rules constituting such expressions make that exceedingly unlikely, as they must ultimately be in some sense useful. But what is useful doesn’t eo ipso translate into “objective existence”. It seems to me that the worry results from that old distinction between the natural and the conventional. And the failure to appreciate how deeply the “conventional” penetrates into the “natural”, (and how bootless, in this age of pervasive technological manipulation, appeals to the “natural” are). As well as, a basic fear of contingency. The worry simply doesn’t occur at the level stipulated and is misplaced. And, on the other hand, it lends itself to such enormous reifications such as, “the universe is a giant computer program” or “the economy is a giant calculating machine”, that are precisely to be resisted as the basis for understanding anything, that reduce understanding to technical manipulation.

Also the question as to whether symbolic “realities” are “mind-independent” or not seems misconceived. Not only does it imply or presuppose some peculiar conception of “mind” and its own independent status, but symbols are precisely “things” that are exchanged communicatively between organisms across the world, and thus are also involved in materially embedded practices that themselves are not necessarily conscious or intentional, (while rendering at least partly the latter possible). Factoring out that mediating “praxiological” realm is what makes for a mistaken binary alternative.

Also it’s curious that the example for an objectively existent function that Bell chooses is “0”, nada. (Though I myself have wondered how they did trade and debt accounting without it, how they registered losses as well as gains. Did they simply just assume that stealing more booty and capturing more slaves would always allow them to “make good”?)

@120:

Probably that pattern recognition, a comparison of more or less, than formally speaking, counting. Monkeys can “count” up to 7. Coincidentally, phone numbers in the U.S. were restricted to 7 digits because experimental psychologists determined that that was the maximum that could be readily remembered.

You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them.

I see your point. And frankly, I don’t have any knock-down argument against constructivism. But I do think I wasn’t really asking whether we have to stick to the rules—rather pointing out that the rules in question seem inescapable. And even there, I wasn’t so much forwarding an argument as articulating what seems a rather strong intuition for a lot of people, one that I myself am inclined to account for in normative vs. naturalistic/psychologistic terms.

Weird formatting error in my last, sorry. There’s a sentence in there that’s utterly out of place; should be obvious which.

98 & 105

I think so. I was thinking of folding a sheet of paper, or perhaps a piece of string. And in Peter T’s link about birds it appears that cormorants can count up to 7, presumably without numbers!

Also, @ 110

The Hamming talk includes fascinating example of deductive reasoning vis-a-vis Galileo and the rate of falling bodies. It also includes this gem, (paraphrasing) “once we move from integers to fractions we have to abandon the idea of the next number.”

126: He said “signs” not “words” right there in the text. Yup there it is “signs”

Yup, and I asked for a clarification. Go f*ck yourself.

“Probably that[s] pattern recognition, a comparison of more or less, than formally speaking, counting.”

A pattern recognition that does comparisons of more or less and recognises “equals” is, to my mind, counting.

@131:

No, Frege’s theory of integers and the like. Establish the interval between 0 and 1, then loop that recursively through to 2 und so weiter. Or something like that. It’s describable as a set of formal operations and infinitely repeatable. Same or different is not equivalent to”logically identical” or equal. It’s an analog process, not a digital one, and likely draws on different mental/neural bases.

Platonic, or psychotic–take your pick:

https://people.maths.ox.ac.uk/~maini/PKM%20publications/252.pdf

Apologies for dumbing down the discussion, but does anybody know of a language that does not have dimensionless numbers? (I am going out on a limb to say that I highly disbelieve that Korean doesn’t have them; Japanese certainly does.) Even the most literal-minded individuals seem able to make use of them, which is interesting.

This is probably of no use for the constructivism / platonism discussion above, of course.

With regard to paraconsistent logics, it seems to me that if you can get “interesting” results, the subset of “useful” results you get are actually obtainable in a more rigorous system – the paraconsistent logic just being useful (I would assume) as a way to short-circuit some of the usual processes (but why?). Also wanted to throw a shout out there for Armando mentioning Godel’s hurdles.

@104 & 139:

I studied logic with Chris Mortensen. He wrote a book, Inconsistent mathematics (Kluwer, 1995). It’s been several years, but as I recall, he demonstrated that you could build an analog to a Robinson arithmetic on a paraconsistent logical foundation. I think the idea is to get away from the inconvenient paradoxes you get with classical logics. As for its “useful” results, Mortensen said you could build a paraconsistent mathematics good enough to kill people with, so there’s that.

This is what kids watch now? Well… at least it’s not the Hasbro GI Joe/Transformers Power Hour.

Let me take a less constructivist/instrumentalist/Platonist standpoint and answer the main question – “are numbers real, or are they just something people made up?” – all Zen-like with another question: Is race real? Is capitalism? Beauty? Injustice? Is green real? I can make pretty solid arguments that those things exist only because of human action and human categorization. Would the world make more sense if you denied those things existed? I think this is way better than doing Frege and Peano because when you go down that route, you end up finding Gödel and concluding that no arithmetic can be provably consistent anyway.

Besides, I wrote an essay once that recast the whole history of the foundations of mathematics in Marxist terms as a succession of class revolutions, moving from feudal-monarchical Platonism to industrial-imperial age Fregean logics, on to Bourbaki’s post-WWI “New Math” for the proletarian “working mathematician”, stagnating for years until a new generation of leadership ushered in a period of glasnost and perestroika after computers took off in the 70s and 80s, and finally leading to modern anti-foundationalism and the 21st century altermondialist computational-algorithmic neofoundationalism of guys like Chaitlin and, in a pinch, Wolfram. I think my version of that history makes at least as much sense as any mathematicians’ treatment, so it’d be hypocritical of me to start rejecting a humanities approach to the problems of mathematics now.

@ 143 — I’d like to read the essay. Otherwise, the question, ‘Whaddaya mean, ‘real’?’ has been raised and the response(s) indeed seem(s) like (a) first step(s) towards clarifying the formalist-realist-whatever debate. (Inflections can be annoying, sometimes.)

Anarcissie@145, I have a chance at a job that would maybe let me wrestle it into publishable shape and I really want to because I think it has the spark to be a very good, if somewhat farcical, essay. If I get this job, it goes into the packet I send to anyone who wants to be my editor.

Consumatopia@146: I don’t actually think that, but most people think I’m an anti-realist about science anyway.

But let me make a Lakotos-Putnam style argument for the sake of devil’s advocacy: Mathematics is quasi-empirical because of its connection to science and technology. Science and technology are real because, well, airplanes and stuff, but seen on its own, math is pure mind games with no reality of any kind. It only takes on any kind of truth when bound up with some application. Even 2 + 2 = 4 is only true because I need to know that my brother isn’t getting more M&M’s from the packet than I am.

Consumatopia@146:

So that we might encounter aliens that have the same theories of physics and chemistry that we do, but it is absolutely impossible to translate between their quantitative system and ours?

We don’t need aliens to follow this example to its logical conclusion. Consider that birds must have some notion of fluid dynamics to fly, or that termites must have some notion of gravity to build mounds. Surely their physical notions must be compatible with ours. They can’t have a fundamentally different science that would let them do magic. But do you expect them to have scientific theories that look anything like that ones I learned in university?

Meant to add: the paper mentioned in 143 sounds really amazing! I do hope it gets published, not least because I’d love to read it.

js@150:

… requires some at least minimal theoretical grasp of what makes abilities like staying afloat

People had aerodynamics before they understood why atoms exclude each other from occupying the same space and I’m not sure the existing answer to that question is anything more than a complex kind of question begging. (Why do atoms exclude each other from occupying the same space? Because they are fermions. Why do fermions exclude each other from the same space? Because they obey the rule that no two particles can have the same quantum state. So why can’t two fermions have the same quantum state? Because if they could do that, they wouldn’t be fermions.) I could imagine an alien saying humans have no understanding of why their airplanes work because we don’t have a complete theory of the universe.

I do think that birds know when they are flying and when they aren’t. I think they have some clear ideas about what air is and how flapping their wings makes them fly. They must have a model of the world and their model of air and motion must be consistent with ours, at least over the range of phenomena a bird is likely to encounter. I’m not sure sure you can define “minimal theoretical grasp” in a way that doesn’t imply “did a bunch of math”, in which case you’ve defined the knowledge it takes to do science as including math as people usually understand it and the claim becomes tautological.

Negative, Captain. Also, how are you using the word “real” here?

Mathematics would be real if when two people disagree as to whether following a set of mathematical axioms leads to a given conclusion, at least one of them is objectively wrong. Grant me just that alone, and the work of mathematicians becomes meaningful, and the application of mathematics to the sciences is justified.

(Mathematics would be more real if there were objective reasons to prefer some axioms to others, and even more real that that if those reasons were capital-N Necessary in some sense (“mind of God”). But I don’t you need either of those to say that mathematics is as real as matter/energy.)

The anti-platonists are saying just because it is useful doesn’t mean it has any existential status comparable to matter or energy.

Math is obviously not a physical object. But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable–it should be possible to describe the laws of physics just as well without math (or anything isomorphic to math). Otherwise, it’s a truth of the physical world that something like math is needed to describe it.

Anon 01.28.14 at 4:57 pm @ 152:
‘… In a way it’s anti-Platonism, since in Plato structures exist not just independently of humans but of the material world, whereas I assume you mean they exist independently in the material world. …’

I think it’s a misfortune that mathematical realism has become conflated with Plato and his works, since the latter come with a lot of (to me) unpleasant baggage. The notion that some sort of higher realm of forms both exists completely independently of the material world and at the same time produces it, ‘a spume that plays upon a ghostly paradigm of things’, is clearly self-contradictory: if A produces B, A is not independent of (disconnected from) B. Just so, the sense that when I do mathematics I am discovering rather than constructing or inventing something, leads me to believe not that the mathematics is separate from the material world, but that they are two aspects of one thing, which become separately visible because of the way they are looked at.

@ 155

Mathematics would be real if when two people disagree as to whether following a set of mathematical axioms leads to a given conclusion, at least one of them is objectively wrong.

“Objectively” is the troublesome word here. Since math is a collection of rules, a wrong answer in math is “conventionally” wrong. At least I think that is a better approximation.

Math is obviously not a physical object. But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable

Well, clearly, it is dispensable in the sense that we could survive without it. Similarly, the world doesn’t need to be described. Plants and animals can thrive without telling stories about the world. Math does an excellent job of describing the world. No coincidence since that is what we devised it for. But we don’t really know how accurate our mathematical descriptions are. If history is any indication we can bet that what is considered mathematically accurate today will someday be discarded as “crude.”

I wouldn’t say that mathematics, of itself, describes anything at all. The conventions come into it, when we want to use math to describe the world. We need conventions to make that possible. Those conventions have to do with making a social agreement on the correspondences to the observable world, which make it possible to communicate a description. The conventions used to describe mathematical concepts and reasoning are of this same kind, social agreements on a correspondence with what is commonly observable. It’s Eve pointing out things to Adam and slapping him alongside the head when he’s obdurate about naming names. But, math isn’t the conventions any more than a rabbit is the name of its kind, “rabbit”.

“Objectively” is the troublesome word here. Since math is a collection of rules, a wrong answer in math is “conventionally” wrong. At least I think that is a better approximation.

No, objectively is definitely the right word (and if you insist on a definition, I’ll pick “not conventionally”.) Math is not a particular collection of rules–see Gödel. You might, as a matter of convention, select a particular group of rules. But it is not the case that because the set of rules you pick is conventional, the truth of whether or not you are following the rules must also be conventional.

But we don’t really know how accurate our mathematical descriptions are. If history is any indication we can bet that what is considered mathematically accurate today will someday be discarded as “crude.”

If the math is somehow “inaccurate”, then our descriptions of the world must also be inaccurate.

Look, it may be the case that all our scientific theories are wrong. It may be the case that there’s some way to replace all of the math in all of our theories with something else that does just as well without being equivalent to math. But those are both empirical questions, and if neither of them is the case, then math is real.

@126 “Are you saying that we can’t think any thoughts for which there are no words? If so, is it an uncontroversial assertion among people who study this stuff? I wouldn’t have thought so.”

Well, there are two ways that people talk about mental behavior. Sometimes by a thought they mean a unit of subjective experience and sometimes they mean a unit of private linguistic behavior. So if you mean the first definition, then you can have thoughts for which there are no words, but if it is the second you can’t.

Since mathematics is a linguistic phenomena, mathematical thoughts must be of the second type, so one cannot think about math without using words. People mix up these two definitions all the time, leading to all kinds of confusions.

But what about the babies and the crows that show numerical behavior but are not able to use language? I think the scientists doing these studies are sloppy with their language, and they should not be saying that babies or crows are “doing math.” I am more of a behaviorist than is fashionable these days, but simple numerical behavior is not at all the same thing as an adult human using language to study math. In the same way, any non-linguistic notion of aerodynamics that birds have is a completely different type of thing than the linguistic subject of aerodynamics that humans use.

If we met aliens and they used language, then we would be able to understand their math, after we succeeded in translating their language. (This basically leads us to Blindsight and thats a whole other discussion.)

The idea that math is made of Ding Dongs also raises existential questions, due to the bankruptcy of the Hostess Corporation. Even watching the video didn’t totally erase that interpretation from my mind.

Bruce Wilder @ 161: I wouldn’t say that mathematics, of itself, describes anything at all. The conventions come into it, when we want to use math to describe the world. We need conventions to make that possible.

5+5=10 doesn’t describe anything, the way “I have five fingers on my left and five fingers on my right and therefore ten fingers altogether” does. But (a+b)+c=a+(b+c) does describe the fact that when adding pebbles to a pile, the order they’re added in doesn’t matter. You’re the first one in this thread who’s interpreted “math” in that way. I believe everyone else who’s referenced “conventions” is referring to things like the use of base 10 numbers rather than base 16 and the use of “+” to mean “plus” rather than “logical OR.”

Peter T 01.28.14 at 12:49 am

@120 Peter T,

“A raven whose brain processes ‘I need friends to help me drive off these three interlopers. I have found one; I need more’ is dealing with a numerical reality, not an articulated symbol system.”

But in this case “three interlopers” isn’t a reality in the sense of a mind independent fact. The total number of interlopers in the universe is a mind independent fact, while the raven’s “three” exists only in relation to the raven. “Three interlopers” means three of them *here*, in relation to this raven.

This is why it’s also extraordinarily strange to say we “discover” zero or that zero is “real.” I hate to side with Sartre on anything, but consciousness brings nothingness into the world; something is only absent in relation to the intention of a consciousness that seeks it. There is no nothing.

I don’t recall Sartre’s specific example, but a true statement such as “Jean-Paul is not at the cafe” is not a mind-independent reality. Jean-Paul’s not-being-there is true only in relation to a “here” of consciousness. The mind-independent reality of this “not” is his being at the next cafe down the street. The reality of “I have zero apples” is that there are apples. There are no zero apples.

But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable–it should be possible to describe the laws of physics just as well without math (or anything isomorphic to math). Otherwise, it’s a truth of the physical world that something like math is needed to describe it.

But that wouldn’t give mathematics any special ontological privileges. I think of it this way: the world behaves in a highly regular fashion. If you begin with that conception, then the first thing you do when you really start trying to understand how it works is to figure out what the regularities are. But the language of mathematics is perfect for describing regularities; you could write out something like the concept of a derivative in natural language if you like, but once you formalize it, you really have a nice regular framework that’s widely applicable. Then it turns out that lots of things in the world behave in such a way that they can be said to obey derivative-like relationships so naturally you take that function or structure or what have you and start using it wherever it looks useful. It seems to me that any coherent description of regularity would in some sense be isomorphic to a mathematical language, if only because you’re really concerned with creating structures that behave in the same consistent fashion that (you imagine) the world behaves, and mathematics is an excellent tool for that.

I’m not sure why you need a computer here. I can do things on a piece of paper, so in that sense they are real. This is an extremely low bar for reality, by the way. That is, if I can conceive it and write it down, then it is real. (I’d note that you wouldn’t apply that standard to fictional constructs. That is, you wouldn’t say that Narnia was real, just because you can write it down describe it etc.)

The computer has one big difference compared you with your piece of paper–it’s behavior can be simply predicted. What matters is not the symbols, but the behavior of the symbols. I can write down a definition of a super-turing oracle, but I can’t predict what it would do, because it’s not real. (If I could predict it, then I would by definition have a super-turing oracle, and therefore it would be real.)

My silly plus can easily be written into a set of rules – mathematicians do this sort of thing all the time. It is really well understood how you can do this arbitrarily. It is also quite well understood that most mathematics you get by writing down random rules doesn’t really go anywhere, but thats by the by.

It’s not by the by, it’s the entire point–any set of rules can be arbitrarily selected and written down, therefore the real business of mathematics is not arbitrarily selecting a pair of rules and writing them down. If mathematics were just a matter of convention (the anti-realist position), you would go just as far with any set of rules.

But sure, if you are saying that any set of rules you write down is real, because you have written them down (despite the fact that different rules are contradictory) then fine.

Any set of rules that you can completely describe is a real rule. That a rule implies a contradiction is a real fact. The implications of the rule are not necessarily true, but that the rule implied them is.

This is about a million light years away from Platonism though, and is really the same position as the anti-realists.

Formalism is much closer to what most people who call themselves mathematical platonists would call platonism than it is to, for example, the Lakoff/Núñez claims about mathematics as a metaphor. A formalist doesn’t care whether babies “naturally think logarithmically”–it’s just a matter of sticking to whatever the defined rules are. The anti-realist would deny that there’s any sense in “sticking to the rules” other than what the neurons in our human brains do when stimulated by the image of the rules on on paper.

Ed Herdman: “Apologies for dumbing down the discussion, but does anybody know of a language that does not have dimensionless numbers?”

Not sure, but how about Melanesian Pidgin? Wanpela, tupela, tripela.

I guess if I say “derivatives are real”, that’s equivalent to “lots of things in the world behave in such a way that they can be said to obey derivative-like relationships”. It doesn’t matter if those derivatives are described by natural language or funny symbols.

That seems like a very pragmatic outlook to me, and one that I would agree with, but I don’t think it’s what people have in mind when they talk about mathematical Platonism. Those claims are usually much stronger, e.g. “Derivative [or numbers or whatever] are actual ontological features of the world,” not merely “useful descriptive formalisms.”

@179, It may not be full Platonism, but it makes two claims that seem to be in dispute: that the term “derivative” refers to a single thing whether it’s described in symbols, language, or some alien communication system; and that in this world there are human-independent facts related to derivatives just as there are facts related to matter and energy.

Peter T wrote:

A raven whose brain processes “I need friends to help me drive off these three interlopers. I have found one; I need more” is dealing with a numerical reality, not an articulated symbol system.

Is this, what you are thinking of?

A H 114: “Given that thoughts are constructed out of an arbitrary system of representative signs, what could conceivably physiologically constrain what those signs could represent?” We don’t have a good account of how thoughts are represented in nervous systems – wait we don’t even have a good account of what thoughts ARE – so how can we know how limited we are in our ability to think? I don’t know and neither do you but it’s a question worth considering.

Scott 149: “Consider that birds must have some notion of fluid dynamics to fly”
Do I have to have a notion of electrodynamics when I switch on the light? Maybe more to the point: Do children have to have a notion of grammar when they learn speaking? Maybe you are saying that the birds (or the children’s) nervous system must encode something equivalent to a physical (or logical) theory and this representation must be accurate enough to work in the real world. Maybe this representation IS mathematics???

@ 163

Math is not a particular collection of rules–see Gödel.

What is math then?

You might, as a matter of convention, select a particular group of rules. But it is not the case that because the set of rules you pick is conventional, the truth of whether or not you are following the rules must also be conventional.

Well, as Armando has said, maybe “formal” is a better way to understand mathematical truth. But, seems to me you’re incorrect. If you make a mistake while trying to follow the rules then you have made a formal or conventional mistake.

If the math is somehow “inaccurate”, then our descriptions of the world must also be inaccurate. … It may be the case that there’s some way to replace all of the math in all of our theories with something else that does just as well without being equivalent to math.

AFAIK we do NOT have a tight mathematical model for the universe. Far from it. So there is plenty of room for improvement. But I was not suggesting that our “math” is faulty, only that there is a good chance that new mathematical ideas and systems will eventually be used to describe difficult-to-model phenomena, of which there is plenty. And when that happens what does that imply about the earlier math? Are you denying that Newtonian physics (and the math associated with it) was shown by Einstein to be only an approximation of a subtler reality?

@185 If thoughts are defined linguistically, and mathematical thoughts should be, then we do know that thoughts are made up of an arbitrary system of representative signs, i.e. language.

@186 Besides a few diagrams, math is just words. When you see an equation on the page you can speak it out loud.

Really guys, there are some overlooked or missing distinctions here, causing mistakes and confusions. Know-how/know-that. Analog/digital. Constitutive/regulative rules.

@192:

There’s a Walter Benjamin line to the effect that painting is the mute language of things raised up to a higher, but still silent language.

Consumatopia,

You can claim that the set of rules you pick is just a matter of convention.

I think I see where you and I are not connecting. So… I never said that mathematical truth is just a matter of convention. Of course not! What I did say was that we ‘devised mathematics to describe the world.’ Not arbitrary at all. We created this language of mathematics for the express purpose of being useful.

But, similar to regular language, it is still a matter of convention that x+y=z. Same with “roses are red.” So I think you overstated my position quite a bit.

On the contrary, that’s my point–assuming Einstein was right, it is a fact of the physical world that his math fits the world closer than Newton’s math.

Wait. I thought it was my point! Our mathematical formulations are approximations which–we hope–are approaching reality. But, they are not reality! They approach reality as a limit. Are you sure you want to call an approximation “real” or “true”?

My belief is just that when people apply the process typically called “math”, there is something human-independent about that activity, at least some of the time.

And I see no justification for such a belief.

Anon @33 had a good point: I can’t for the life of me convince myself that the question “are numbers real?” has a clear meaning or whether the commitment either way leads to interesting further commitments or consequences … What are the stakes of such a debate? What do we lose if numbers aren’t real?

Have been reading along hopefully, but I don’t see that anyone has answered, or even addressed his questions. (Well, maybe jch.)

In the same spirit, I’d ask: do we have an agreed-upon meaning for “exist”? One of my dictionaries defines it as “to have being or actuality of a material or spiritual nature.” The other defines it as “to be; to have reality or actual being.” As far as I can tell, these definitions are circular, hence utterly useless.

When I was a devout Catholic long ago, I made the mistake of beginning to study Scholastic philosophy. I was stopped cold by “Being.” I couldn’t make any sense of the notion, as metaphysicians use it. Then I came across the logical positivist/pragmatist dictum: “Faced with a metaphysical question, ask: What depends on the answer? How would the world be different if one answer or the other were true?” For the (eternal) life of me, I couldn’t see how the world would be different if “essences” or “substances” or “souls” or “numbers” existed in (metaphysical) reality, rather than as counters in a language game. I still can’t.

Link should be this: http://plato.stanford.edu/entries/wittgenstein/#Rul

mattski:

If you hold that mathematical expressions are (more or less) just like other nominative expressions, you’re either going to have to hold a radically anti-realist line, or you’re going to pretty much concede something very much like Platonism. “Roses are red” is a complicated example, but take “gold is a metal”. Obviously, the signs ‘gold’ and ‘metal’ depend on certain conventions, but we use to pick out these conventionally defined signs to pick out aspects of independently existing reality. So, gold e.g. is an independently existing kind of thing that we refer to using the conventionally defined symbol ‘gold’. Now suppose for the sake of argument that our category of ‘metal’ is inexact and will in the future be replaced by a more exact category—this future category will be more exact because it’ll more accurately pick out the independently existing kind of thing that we now refer to using ‘metal’.

Now if mathematical expressions are just like that, then ‘2’ functions just like ‘gold’ or ‘metal’ does—it refers to aspects of independently existing reality. Of course, ‘2’, ‘two’, etc. are conventionally defined, and of course they refer, if they do, to abstract entities and not to material objects or kinds of objects, (tho kinds may be abstract entities anyway, but just skip that), and moreover it won’t matter if ‘2’ etc. are inexact somehow and will be replaced by expressions that do a better of job of categorizing whatever it is that we use numerical expressions to pick out now.

This isn’t really my view, by the way. It’s just that if there is an important distinction between the way mathematical expressions function and the way in which expressions designating spatiotemporal objects function, (as I do think there is), then this distinction can’t turn on how exact or approximate these expressions are.

So we’re all just shopping in Belle Waring’s “Philosophy Museum and Antique Curios Emporium”? Get your transcendental objects here, folks! Empirically quite cheap!

Mathematics depends on the SAME cognitive operations as physics and chemistry: Seeing things particulate or separated, carrying a memory of one to the next, finding connections between them, etc. The necessary operation is that the lifeworld splits into two (Plotinus and LEJ Brouwer).

Humans could have math but with a very different physics and chemistry. In an alternate universe. If an alternate universe might have a different set of physical constants (as we now suppose is possible), then why not a different set of physics equations? Go the whole hog!

But an alien race, even in this universe, which exists as colored lights that hum to each other might not have our metaphysics of seeing particles in forcefields. They might not have the necessary intra-cognitive splitting. They could be extraordinarily advanced in understanding the universe, but they might not do math, physics or chemistry.

Perhaps stars like our Sun are conscious. They only feel other stars, they are only conscious to other stars. (They don’t bother with their planets; we are motes in their eyes!) In all their lightbeams, time is stopped dead (that is OUR physics) — thus, they communicate instantaneously. All stars in the universe are in instantaneous communication; maybe they compose one big creature. Okay then: It is grander than you. Does it do math or physics?

Whales have bigger brains than humans, and appear (to me) to be more intelligent. There is evidence that their language is complex and memorous (or memory-laden). Do numbers exist for whales?

Please carry on.

We just don’t know what the neural basis of subjective consciousness, willful action, or language is.
Not just that, our designated thinkers-about-these-things don’t seem to have picked up a lot of humility on the way to that not knowing.
A hundred plus years max of even the bare ground of microbial awareness and where’s the “Wow! We thought we knew pretty much what was up on the local scene and it turns out an invisible reality was saturating our lives all along, and we had no idea it was there.
Here.”

Didn’t even know that what we are ourselves to start with is just another tiny thing on that microscopic scale.
And the whole time it was that way and the whole time until just yesterday we didn’t know anything about it except how to make beer and bread.
So what’s the odds on more to come? At that level of shift. Of awareness.
What’s the likelihood that if there is as Lee Arnold suggests, a kind of sentience in the stellar nucleus, it’s so far beyond our grasp it might as well not be there.
Not someone you could converse with, or make demands of, but exactly what you get when energy of that magnitude exists in one place, which it does.

I can’t separate wheat from chaff in sci-journalism, but Sir Roger Penrose is evidently pretty chuffed about being right about something microtubules quantum something brain. Thought.

Up and down don’t really exist. They’re just a local convenience. Like numbers.
Vital, essential, and they disappear completely around ten miles up, as well as just over the horizon.
Not to mention the non-theoretical center of the earth. Try to get “down” working there.

It’s hard for some of us to imagine reality without mathematical essence, it’s like describing “a box with no inside” or making a drawing of a Klein bottle.
A word game, not a weirdness at the heart of things. Or a glory.
But then some people can’t imagine thinking without words, or signs. And others can’t imagine not being able to do that.

Is the weird kid at school doing mathematics when she annoys her teacher by getting the correct answer while being unable to explain how? I’m not sure that a purely language-based system encompasses all of what I would want to include.

Re: the Hellenistic tradition, I think I’m right in saying that they had a concept of number that was restricted to integers. Rational numbers would have been considered as something else (ratios but not numbers), and irrationals simply as magnitudes, again strictly not numbers. And of course they did lots of interesting and correct mathematics, just not all the kinds of things we can do using a broader definition. (I’m using ‘we’ in a broad sense obviously – I’m hopeless).

Lee Arnold at 4:59 am – that was really interesting (your posts are often mind expanding for me). Care to elaborate on whale intelligence? Thanks.

Yama #213 — Thanks for the vote of confidence but I am afraid that if I had any intelligence at all I’d be studying cetacean communication in some place like Hawaii!

should I express mild surprise for having successfully trolled everyone with alleged doctrinaire adherence to the actual Platonism of Plato in particular?

I think people were afraid John would pop in and tell them they didn’t really believe what they thought they did.

Actually, the rules of implication are up for grabs. For reals. Not only could you see the axiom of choice like this – what mathematical objects can we work with – but also more fundamentally in things like accepting or rejecting the law of the excluded middle.

You can use a constructive logic that doesn’t hold either of those, but still derive statements conditional on them being true. Better examples would be substructural logics, in which weakening, contraction and/or exchange rules are not permitted. (If you’re using linear logic without exponentials, you can’t put something on the left of an implication that lets you do unlimited contractions on the right.) Still, even if different logics derive different statements, there should still be a fact of the matter as to which statements are derivable in each logic (if there isn’t then we have to take exactly the anti-realist position you claim no one is taking).

It is also important to note that if you execute a set of rules on a computer – eg arithmetic – then there will be things about those objects which the computer cannot see. If this is one’s criterion for reality, would that mean that there are facts about integers which are neither true nor false?

I don’t know if I’d go so far as to say that only semidecidable properties have a “fact of the matter”, but the further you climb up the arithmetical hierarchy the more in doubt that is. I think it’s exactly that kind of doubt that makes it reasonable to talk about whether a mathematical truth “exists” or not–not all of them necessarily do.

It demonstrates a reality that no one opposes, so doesn’t say very much.

Well, no, I think some people do oppose it. Some people have a much more radical anti-realist position than you do. I was badly mistaken in assigning that view to mattski, and I guess there’s doubt as to whether even late Wittgenstein believed that (I’m no expert), but I still stand by the statement that this is not a strawman, even if I’m not sure exactly who’s putting it forward ;-).

That aside, I think the possibility of implementing rules in machines has more relevance to the question of whether math is discovered than art is discovered because mathematics is inherently about the behavior of rules, while Casablanca is not at all about the behavior of video decompression algorithms (unless I misunderstood it completely!)

I’m pretty sure it is a strawman, Consumatopia, and that the position you are describing as “realist” is more commonly known as “anti-realist”. This makes the exchange with you a bit odd, to say the least. But, in any case, we agree that one can write down the rules for a formal system and more or less decide certain facts about it.

I’ve enjoyed the pastime of seeing what meta mash people will try to make of mathematics, but in some ways I am left wanting more.

The power of math is a bit mysterious, no? Does it depend on notation? I think it kind of does, but why is that so? If notation is tied directly to human cognitive incapacities, does that shape math itself? Is notation neutral? Neutral in relation to what?

Is the demand that math be “real” just a tribute to its surprising power? The kids think nothing is real until it is on teevee, maybe math isn’t real until it’s on YouTube.

And, finally, let me reiterate my previous complaint: too many are way to eager to leap to an uncritical assertion about math being “descriptive”. What is the relation of math to poetry? Or drama? Or comedy? Or rhetoric?

This, by the way, is what contemporary platonism looks like. It’s not really that weird, and it can incorporate worries about approximation and maybe even intuitive logic (i.e. denial of bivalence), tho this is possibly problematic.

I think it would have been more fun to have indoctrinated your kids into logicism, comme ça:

http://isites.harvard.edu/fs/docs/icb.topic1240846.files/Benacerraf.pdf

js.

Apologies. I would like to respond to you in a more considered way but today is shot. So just this:

If you hold that mathematical expressions are (more or less) just like other nominative expressions, you’re either going to have to hold a radically anti-realist line, or you’re going to pretty much concede something very much like Platonism.

I don’t understand what you are trying to get across. I don’t understand what position you are imputing to me.

If the numbers are in some meaningful sense outside of us, by what means do we interact with them, the pineal gland? Pencils?

But they aren’t *only* outside of us, in (my version of) this view; they are inherent in everything, including whatever we happen to be thinking with, and whatever squid or aliens or pure asensory AIs think with. They’re here for us to feel out, although we (and, I hope, squid and aliens etc etc) will feel out different aspects.

I don’t think this makes me a panpsychic, but I can go along with `Every physical entity is doing a computation.’ Sadly, some of them seem to be doing it faster than the simulation I’m trying to code.

William Kahan’s advice is to never design a program on your best day, because you need your best day to test and debug it. I don’t know that I could work that up to a theology, but certainly there’s a good origin tale or so in it, with or without panpsychism.

“Mathematics is useful because something like the Strong Church-Turing thesis”

Ugh, that’s not what I meant to say. I mean that if SCT is true, mathematical reasoning will be useful.

Wikipedia paraphrasing Whitehead on the fallacy of reification:

He rejects the notion that a concrete physical object in the universe can be ascribed a simple spatial or temporal extension, that is, without reference of its relations to other spatial or temporal extensions.

Well, okay, but there are relations between objects, so we’re good — no, apparently they aren’t good enough to build on. Why not?… The definition of extension (why not `extent’?) takes us to the Planck length and Spinoza, but not the initiation of the Big Bang, which seems like a missed opportunity.

You can modify Church-Turing to permit randomness (http://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis#Variations ). I agree it kinda seems like you would have to do that for SCT, but not everyone does (http://en.wikipedia.org/wiki/Digital_physics#History )

John and I agreed that each view is open to as many fatal objections as the others.

Belle,

What you and John are looking for is the synthetic a priori. You might think I’m kidding. And yet.

(Ok, maybe I’m half kidding. A third kidding?)

Replacing the Sacred Integers with Sacred Ratios just makes you a Pythagorean, and that’s even worse than a Platonist ;).

Switching to Cellular Automata makes you a Wolframite. Worse still!

@248 — John you are probably right, because I always make the mistake of thinking that “panpsychism” includes developed varieties such as Leibnitz’ “monads” and Whitehead’s “actualities” with vectorial “prehensions” — things which (to me) skirt the borders of consciousness. I suppose that is wrong, and panpsychism should be kept as the simple label for “thoughts of rocks”?

And you are certainly right, insofar as Whitehead was a theist, and God puts Whitehead’s process-metaphysics into motion. So I imagine that Whitehead would have taken a very strong objection to hearing his philosophy called “panpsychism”.

@ 230

I wrote, “A scientist, of all people, is more likely than most to allow that it is possible for a rock to roll UP a hill.”

TM responded, But that position is itself based on mathematical theory (namely, statistics). It actually expresses confidence that current math-based physical theory is a good description of reality.

First of all, it isn’t based solely on theory. It is also based on an attitude of agnosticism about what is and isn’t possible. You could say, the ability to refrain from making absolutist statements.

Second, I don’t see how statistical means of understanding the physical world strengthens the case for mathematical platonism. Especially since fairly simple visual means are available to make statistical statements without mathematics.

Really, statistics strikes me as a very modest form ‘knowledge.’ And from what we know (?!) about quantum mechanics the physical world rests on a foundation of scarcely knowable chaos. Or so it seems to me.

TM We could perhaps imagine a Universe without consistency and not only would there be no Math

Suppose that just this second the Universe began behaving somewhat inconsistently.

Has any mathematical proof thereby become invalid? Will prime numbers suddenly find themselves commonly composite (though I suppose there are just as many of each)?

“Suppose that just this second the Universe began behaving somewhat inconsistently.”

Nice question. But now we have to specify what is meant by inconsistent. My argument would probably have to lead me to say that inconsistency in the sense I mean the term would preclude our continued existence, so nobody would be here to worry about prime numbers.

mattski: I guess we don’t agree on terminology. I don’t think we could have pie charts and not have mathematics.
“So, only a person who believes this specific model can also believe that a rock could possibly roll up hill?” Nah, that’s nothing to do with my argument. I think there’s a difference between saying that A is predicted to be possible by scientific theory X, and A could happen because of magic. Again, we may disagree that these positions are different.

engels 231: I always wonder when reading articles like that: what happened next? Did somebody come up with a refutation or was it just left hanging there?

I don’t understand what you are trying to get across. I don’t understand what position you are imputing to me.

mattski,

I wasn’t so much imputing a position to you as presenting you with a dilemma (sort of). Briefly: if you want to say that (a) numbers and math more generally is a set of heuristic devices we’ve invented because it helps us explain certain aspects of the world, and that (b) things metals, roses, etc., do indeed exist in the world independently of anything going on with us humans, then you don’t want to say that (c) our ‘math language’ let’s say functions more or less just like the language we use to designate spatiotemporal objects—or that out ‘math language’ and our ‘object language’ is conventional in the same way. Asserting (a), (b), and (c) together can quickly get one into trouble that needs a lot of fancy footwork to avoid.

Two other things:

1. Platonists obviously don’t think that numbers, e.g., are spatiotemporal entities! So asking “where” they would be is quite besides the point. Rather, the claim is that we can and should include certain kinds of abstract entities in our ontology—since we’re talking about abstract entities they’ll of course lack spatiotemporal properties like location. Now you might think that this is weird enough (and I e.g. think that the epistemic problem Belle mentions is pretty fatal to the view), but you don’t want to strawman the platonist.

2. jch’s 234 is q

As far as I can tell…

The intuitive pull of Platonism for some derives from what happens when we ask questions like, “if no one were around to add 2 and 2, would 2 still exist as a number and would 2+2 still be equivalent to 4”?

And what happens, of course, is that we view the world as if no one is there – but of course we’re the ones viewing it – so we’re there. We can’t imagine a world without us, in some sense.

Nonetheless, when we think we are imagining a world without us, then it seems perfectly plausible that 2+2=4 even when all human beings have turned to dust. After all, when we ask whether 2 would still exist in such a world, we’re not looking for it visually – we’re really asking whether it makes sense to think of the number 2 in such a world. And it does of course, because we’re doing the thinking.

But frankly I’m not clear on what an abstract object is, or for that matter what a thought really is. I don’t think anyone is clear on those things. And I think you’d need to be clear on those things to settle on what mathematical Platonism is about (if anything at all).

How about this question:

You happen to be there as a badly damaged spacecraft skims to a landing on a frozen river in northern Maine. As it coasts to a stop, the hull splits open, and a creature resembling an odd mix of dolphin and human staggers out, his uniform ripped and bloodied. In one hand, the gray skin blistered with burns, he clutches what appears to be a book. Collapsing into the snow in front of you, he hoarsely tells you in perfect Icelandic that he is the last of his crew and his species; and that the book contains the history of his species and an important warning. But, he gasps, the language is very unlike any human language, and is heavily encrypted as well. Before you can make any inquiries, he places the enormous book into your gloved hands, and dies.

Curious, you open the book, and find remarkably that whenever your glance falls upon a series of symbols not separated from one another by a space, the book emits a certain collection of sounds, some barely audible.

Do any concepts contained in the book that are unknown to human beings exist at this moment? Do they exist only if decryption and translation are logically possible? Do they exist only if decryption and translation are actually possible? Do they exist only if decryption and translation are actually effected?

If one views mathematics as a description of the physical world, then there are some similarities between those questions and the one in this thread.

TM,

I don’t think we could have pie charts and not have mathematics.

How about a picture of several apples, some of one color and some of a different color?

Nah, that’s nothing to do with my argument.

I get the feeling you are indifferent to the context of my original statement. But I’ll offer you this, if an event occurred which seemed to violate our current “laws” of nature–a rock rolling up hill to take an unlikely example–you can bet that scientists would look for an explanation while suspending judgement. Magic would not be invoked.

“How about a picture of several apples, some of one color and some of a different color?”

In a specific sense, this IS mathematics, or let’s say concepts that we think of as mathematical (set, number) are implicit in that picture. I’m still not sure what your point is though. It seems to me that your position is sort of conventionalist and you think I am platonist but I am actually realist. I see math neither as existing independently of empirical reality nor as something we make up.

I said above that for mathematics to be conceivable, the Universe must be “consistent” in some nontrivial sense. To spell that out a bit, a minimum requirement for that consistency is that there are physical entities that can be in distinct states, and that there is some regularity in the way these states change. That for example is a minimum requirement for information to be physically representable and transmittable. Without that, there could be no life, let alone intelligence. Now as soon as there is a system capable of representing distinguishable states, the primitive set concept arises. In that sense, “set” is an aspect of physical reality. An intelligent being trying to simulate (i. e. mentally represent) physical entities cannot do it otherwise than by employing the concept we call set – whether or not they come up with a language for that, and whether or not this happens in some sense consciously.

But physical reality doesn’t contain an entity that is a “set”. It contains entities with distinct states that can be represented as sets. Likewise numbers, and so on.

@267:

Dear Belle Waring,

What makes you think that comment was made in reference to you?

In a specific sense, this IS mathematics, or let’s say concepts that we think of as mathematical (set, number) are implicit in that picture.

-A picture of many apples
-A picture of a large circle next to a small circle
-A picture of a small circle inside a large circle
-A bowling ball placed on a table next to a golf ball.
-A table upon which are placed many apples.

Are you saying these are “mathematics?” If so, I think you are speaking your own private language. There is also the sense that you’re jerking my chain. I hope that isn’t the case.

and you couldn’t perform that description without a set concept represented somewhere in your nervous system

Do you have evidence for this assertion?

And to say that by describing apples on a table you are employing mathematical concepts is not an abuse of language

That would depend heavily on the nature of the description, if you ask me. (Leave aside that I was pointing here at scenarios rather than descriptions of scenarios.) But I don’t see how you write what you have written after saying @ 256 that you “weren’t defending platonism” unless you niggle over definitions to the point of absurdity.

I’ll stop here.

Palaontological taxonomy definitely uses a lot of math.

The images described in 272 are very likely used in (early grade) math textbooks.

And, in the birds-do-fluid-dynamics worldview, the air is also calculating; the bird and air are calculating together.

When is `reification’ self-descriptive?

If I hold to a metaphysical “-ism”, it is agnosticism. The easy way out! But really I don’t trust the arguments and judgments of people who insist that there IS a God, and I don’t trust the arguments and judgments of people who insist that there IS NO God. Rationality could not supply the answer either way, and there’s the end to it. Mathematical Platonism is much more interesting, but still I am agnostic, because how can you possibly know?

That said, I think mathematical Platonism is very easy to defend:

Causal processes have been investigated and sometimes it is found that quantities are conserved. To find that out, you have to count or measure. The number “2” is a placeholder for a position in the act of counting. Its meaning is different from “1” or “3”; “2” is after “1” and before “3”. But to say that numbers are therefore merely the placeholders in the communication of a measurement, in order to avoid saying that numbers exist Platonically, advances you nothing. You have already given that the world has real causal processes, even if no humans existed, and I am going to assume that you will allow that some quantities are conserved in some of those causal processes, even if no humans existed. So quantity exists Platonically. Quantity is number.

Hey jch, here’s an example of reification I think you might like.

The Maxwell’s demon thought experiment–a tiny entity that sorts individual molecules to reduce entropy–was originally conceived to illustrate the statistical nature of the Second Law of thermodynamics. But then people started to ask exactly why such a tiny organism or machine would have to violate the laws of physics at the atomic scale. One suggestion: the entity would have to expend energy (tiny light signals were imagined) to measure the surrounding molecules in order to sort them, and this energy would have to be at least equal to the energy gained by sorting. That’s wrong, but a lot of people believed that (perhaps because people are too quick to want a paradox resolved, by analogy with an observation under the Uncertain Principle, or by analogy with then-popular Cybernetics in which information was inherently valuable (though I think that Norbert Wiener himself actually avoided this mistake) and decades later in the late 90s that’s still how the problem Maxwell’s Demon and it’s resolution were explained to me by very smart people. I’m sure a lot of people still believe it today.

It turns out that the laws of physics, as believe both back then and today, allow a mechanism to copy information (to measure some part of a molecule’s state, put another molecule in correlation with that state) with arbitrarily small increase in entropy. What stops Maxwell’s Demon from breaking the Second Law is Landauer’s principle: resetting one bit of information (dividing the possible number of microstates in the macrostate of the world by two) must increase entropy by at least kT ln 2, which is the most that the Demon could increase energy in any cycle. The time reversibility of the laws of physics (at the micro level, not at the level of daily experience) forces us to pay a price for forgetting information, not for learning it. (Whether or not that’s how reality works, the notable thing is that it’s how the theory works, and people were getting that wrong.)

It’s my suspicion that mistakes somewhat like this are what’s actually at the root of some of the more disastrous social failures of applied abstraction (not just the Wall Street quants, but everything in Seeing like a State). But I don’t have a full theory yet to back that up.

But here my disagreements begin. I would not accept your solution to this problem–that we never look for patterns across disparate domains. In particular, I think that the way we keep making mistakes across all these domains in similar ways probably means that there are generally applicable things to learn here. But I don’t think we’ve precisely explained what those ways are. It’s not just “Reification!”. Not every thought process that someone chooses to label reification is fallacious. Some of this is perfectly valid.

I don’t think the right lessons from Wall Street and the disasters enumerated in Seeing is “destroy all abstraction!” Better responses: beware of technocrats. Ban high-frequency trading. Increase capital requirements for banks. And dsquared is probably right that some kind of crisis was inevitable given monetary and fiscal policies of the major countries. And the finance-physicist apologism of The Physics of Wall-Street wasn’t totally convincing, but there were some good points there. Platonists might not even be the droids you’re looking for here. The Black Swan frequently claimed that Platonists and Bayesians had ruined everything–if only we’d listened to Mandelbrot! Thing is, I’m pretty sure that most platonists are more fond of Mandelbrot than Bayes.

More generally my view is don’t stop people from experimenting with radical ideas, stop people from ruining innocent lives with those experiments. You’ve given one side of the stakes. But not the other. A lot of physicists and mathematicians clearly have platonist inspirations–even if platonism is wrong, they might not have made the breakthroughs they did without believing it (or gone on to study the fields they studied at all). Gödel probably being the most prominent example of that. There are entire research programs that are called into question if mathematical anti-realism is true. (e.g. meta-mathematics and mathematical quasi-empiricism). I happen to think those programs look extremely promising, and it would take much more that the arguments you’ve presented here to convince me otherwise. I did not lack “any conception of reification”, nor do I think you’ve provided much evidence that anyone else in particular does. Just because someone disagrees with you and/or Whitehead doesn’t mean they haven’t heard of this stuff at all. You can conclude that we’re wrong, but you’ve got no basis to attribute as much ignorance as you did.

And the claim that physics is the one real and true scientific account of the world and makes the world

You definitely don’t have to make any reductive claim like that to be a mathematical realist or platonist of any sort. David Deutsch is platonist perhaps even more extreme than Tegmark, and he’s argued at length against a “theory of everything” (not necessarily the attempts to reconcile QM and relativity, but the reductivism implied by the phrase ). Physical reductionism underlies some of the mathematical anti-realist positions (see Lakoff and Núñez linked above). I find it a bit odd that you would accuse platonists, of all people, of having a “tendency toward sheer instrumentalism” or denying ordinary experience.

To repeat myself tediously, mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses. and some legitimate branches of natural science make little use of such formalisms, (e.g paleontological taxonomy). So the argument that because math is used in science and science is referent to reality, math is intrinsic to the universe somehow and thus “ontologically” real, isn’t warranted.

No one has to deny that mathematical formalism are conceived for reasons other than modeling an empirical theory, that there is much math with little apparent value to science, or that some fields of science have more math than others, to argue for the “unreasonable effectiveness” of math in some sciences, or for mathematical realism or empiricism in general. In fact, I would say that math being useful some but not all of the time suggests that it’s apparent utility is not just a matter of everything looking like a nail when I hold a hammer.

Insisting that their “ontological” status is that of reality, (independent of the very complex rules systems that constitute them), just strikes me as odd.

The Löwenheim–Skolem theorem says that no formal theory of an infinitely sized mathematical model can fully specify which model its referring to–there are always some annoying non-standard models (like a set of natural numbers, but where some “number” is its own successor) sneaking in. Slightly more prosaically, by Gödel, any formal theory of mathematics must include some true statements that cannot be proven in the theory. I don’t expect that stuff to convince everyone, but maybe it might help you understand why it doesn’t strike everyone as oddly as it strikes you? To put it another way, what part of “Incompleteness” don’t you understand–the formalisms aren’t complete because there is always more math outside the formalism. (And it isn’t because mathematics is somehow “too big” to describe in a single formal system–it’s because doing so generates a contradiction.)

Anyway, sorry for the mega-comment.

“But then why is John Holbo replying to my reply to Belle Waring’s comment that was addressed to me, other than for the sake of sheer bumptiousness?”

It’s perfectly normal to point out to people that they are being a bit rude, John. And it’s perfectly normal to point out to people that they are being a bit unreasonable. I don’t deny a motive of self-amusement. (Bumptiousness is not my style, however, as you well know.)

Let’s try to turn an etiquette point into a recipe for improved philosophy. Your contributions to this thread are no doubt thoughtful and serious. Thank you for that. But your signature style is dogmatic, and perhaps a more dialectical approach will suit the subject better. When someone disagrees with you, don’t just speculate wildly about how they lack basic concepts. If they are educated, and have studied philosophy, it probably isn’t that. Consider whether they might not have a point. Or half a point.

So let me propose an exercise (I do not say I set it for you – I am not your teacher, I know that). Write a forceful rebuttal to your own #275. Don’t fantasize that the opposition is going to make flagrantly stupid assumptions like, “because science offers a unique, privileged access to the underlying reality of the world, scientific concepts are immediately and intuitively referent.” (It isn’t very likely that people are going to fail to see the problem with that, is it?) Instead, see if you can imagine the opposition avoiding the most obvious pitfalls, yet without being driven to accept what you say.

Think about how your own use of ‘concept’ (for example).

Think about how this is going to provide the opposition with several avenues of attack and objection:

“mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses. and some legitimate branches of natural science make little use of such formalisms, (e.g paleontological taxonomy). So the argument that because math is used in science and science is referent to reality, math is intrinsic to the universe somehow and thus “ontologically” real, isn’t warranted.”

This misrepresents the opposition and, what’s more, it encodes presuppositions that are likely to be held against you. In some ways you have their argument upside down and backwards. It is the very fact that math is sometimes done in an utterly non-empirical spirit (as you say) yet is then shown to have some empirical application (as you admit) that drives the ‘unreasonable effectiveness of mathematics’ line. I don’t by any means regard the simple argument along these lines as decisive, but you are not doing justice even to the simplest version. So obviously you aren’t refuting potentially more sophisticated developments of it.

You are bopping from point to point, opening vast fronts that invite skepticism and attack. “mathematical symbolism, which refers to pure formal operations, is not the same as linguistic symbolism and likely involves different thought processes with a different neural substrate”

How do you know that math symbolism (always?) refers to pure formal operations? Do you really regard it as likely that the neurology is so functionally bifurcated as you suggest? (Wouldn’t we have seen it already in the scans?)

You are going to object that this is just a comment box and you can’t fill in all the details, and guard against every misunderstanding. But then: do the opposition the courtesy of assuming the same. They, too, are limited by the comment box format. I’m not going to call you an idiot, just because it looks to me like several of your comments are open to quite elementary objections. Now you return the favor. (Get it?)

Your view is sophisticated and serious, but you don’t take seriously the thought that the opposition is also sophisticated and serious, not merely lacking in basic concepts. Philosophy of mathematics is marked by sophisticated and serious people saying things like what you say, but also by sophisticated and serious people who defend very different views. (It is also marked by sophisticated and serious people seriously straw-manning the opposition. You are in good and respectable company, no doubt. Still, a little of that goes a long way, surely.)

jch, not wanting to step on John Holbo’s toes again because that’s heckuva rude and not the first time I’ve done that, but it is clear that I misread you somewhat and I apologize for that.

I would say, though, that some Platonists physicists and mathematicians I’ve read are quite aware of the possibility of instrumentalist readings of science and very clear that they don’t like it. But I haven’t taken a survey.

I read back through some of this to see if anyone ever said exactly what they meant by ‘real’ or ‘exists’. I think it is the necessity of this definition which brings in the thicket of phenomenology.

Also think I see JCH’s point a bit clearly now with respect to eliminationism/reductivism – the problem apparently not being that we are stuck somewhere along the path of causality (from the foundations of knowledge) but rather that some reifications are totally separated from that path (untestable) while simultaneously appearing plausible.

“I also have probably a more relaxed sense of argument than you”

I dunno, I’m a pretty relaxed guy.

“As for bumptiousness, I thought it was a “truth universally acknowledged” in these part that academic philosophers are peculiarly prone to such a vice, with their “knock-down arguments” and knocking down of arguments and the like.”

Sorry, are you faulting me for not living down to these low standards?

My point was this: given that you hold your own arguments only to a loose and relaxed standard – nothing wrong with that, especially not on a blog – given that you think you have reasons you aren’t offering at the moment – nothing wrong with that – it is intellectually unfair and unhelpful to assume that those on the other side who are also arguing a bit loosely can only be doing so because they lack basic concepts, or are otherwise making some really obvious error. You would not like it if I told you that your relaxed arguments were indicative of lacking basic concepts. So you should extend the courtesy to the other side – not merely out of politeness, but because nothing intellectually good can come of doing otherwise.

Ergh! I forgot to mention the matter of “professional deformation”. And believe it or not, I read and am influenced in my thinking by much else besides philosophy. Nor is “reification” a basic concept; it is a conception of what can happen to “basic concepts”. Nor did I fail to extend “basic courtesy”, (as if that were the sine qua non of all internet traditions). Nor Is “intellectually good” some unique desideratum: there is such a thing as mistaken rigor. And it tends toward meaningless games of intellectual one-upmanship, which are less about “truth” than performance and the will-to-power. Nor did I impugn the different heuristics by which others operate. Rather, certain persons may have *chosen* to take offence, in the name of “courtesy”, for obscure reasons or motives of their own.

Ergh!

“Ergh! I forgot to mention the matter of “professional deformation”.”

Are you taking ‘Holbo is deformed’ as a premise, or a conclusion? Or neither. It makes a bit of a difference.

“Nor is “reification” a basic concept;”

In philosophy, the idea that you might mistake a substantive for a substance is pretty basic. It’s the 21st century, after all.

“Rather, certain persons may have *chosen* to take offence, in the name of “courtesy”, for obscure reasons or motives of their own.”

Let’s just grant that there is a certain amount of will-to-power and one-ups-manship on all sides. The question then becomes: might there be any intellectually sound reason for me to point out that you seem to be straw-manning the opposition? I think there obviously is. (Even if my motive is utterly bogus. I am a professionally deformed one-ups-man. Let it be so.)