Platonism in the Philosophy of Mathematics (2018) | Hacker News Reader

Platonism in the Philosophy of Mathematics (2018)

plato.stanford.edu

89 pointsKhoomeiK119 comments

This is from the Stanford Encyclopedia of Philosophy; see also the corresponding article in the Internet Encyclopedia of Philosophy: https://iep.utm.edu/mathplat/

Also, frequently quoted (by me at least): from an interview with Yuri Manin (http://www.ams.org/notices/200910/rtx091001268p.pdf):

> I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something.

I wonder why he thinks there are no rational arguments in favor of Platonism. I think there have been a lot of convincing ones over the last few thousand years.

Yeah, the physicist Tegmark has arguments for a mathematical multiverse. Sean Carole has a podcast interviewing Justin Clark-Doane who argues that at least arithmetic must be real if the world is to be understood as consistent.

https://www.preposterousuniverse.com/podcast/2021/03/08/137-...

Flat Earth is sort of consistent somewhere too. But platonism isn't an idea on the level "looks like that somewhere", it aspires to be a deeper fundamental view of reality.

Although he may not be convinced by the rational arguments, I took the quote as mostly expressing the emotional/irrational draw of it; the dismissal as hyperbole to support that as a rhetorical device.

You may want to review Russell's A History of Western Philosophy. Bertrand pretty effectively shredded most of the fantastical precepts Platonism relies on in his analysis in the chapter on Theory of Ideas (15 in the Audible audio book).

The basic gist is that the basis of Platonism is metaphysical more than logical. Mysticism is the key bedrock of the form of the Good. Plato's ideal of Ultimate Truth is self-contradictory when reconciled with Geometry. Plato's lack of subtlety with syntax causes a lot of his arguments to fall flat, such as "beauty is beautiful." The analysis of the Parmenides is beautiful, highly recommended.

If your comment doesn't exist, how can we read it?

This is one of my favorite questions in the philosophy of mathematics. A related question for me is the ontology of objects we associate with Software -- algorithms, data structures, etc.

Software is a level of abstraction, but I never saw an evidence that philosophy knows the concept of level of abstraction.

What?!? Russell literally developed Type Theory. Even discussion of Platonic Forms ultimately goes from "lesser" abstractions like the Ideal Form of a Bed to "greater" abstractions like Beauty, Justice, Good.

Please, read more philosophy. Clearer, more informed thinking might lead to better software.

That's emotional level, not structural. When software supervenes on data and code, data and code supervenes on bytes, bytes supervene on bits, bits supervene on transistors, transistors supervene on electrons, electrons supervene on the field.

I highly recommend the book Anathem as it goes into this quite a bit.

I wish my kitchen had the tile pattern that the 1000er completed:)

This topic, slightly more popularized, appeared in the Science Diction podcast: https://www.wnycstudios.org/podcasts/science-diction/article...

When we ask "do they exist", we invite a certain type of reasoning. "Well consider chairs. We can all agree they exist. In what ways are mathematical objects similar and different from chairs?"

But I have an idea. What if we turn it upside down? "Do mathematical objects *not exist*?" Now we invite the opposite metaphor. "Well consider ghosts. We can all agree that ghosts don't exist. In what ways are mathematical objects similar and different from ghosts?"

I think that fundamentally if you say that mathematical objects do exist, then you have to admit basically everything in - including ghosts. Fundamentally, definitions are a matter of taste and aesthetics, and I dislike a definition that is so broad as to contain everything. That makes it useless for reasoning and discussing.

Would you say that there exists something in common between the 21 sheep and the 21 dry peas in the shepherd's pocket which he uses every evening to see if any of the sheep have been lost? That common which exists, in a very clear and practical sense, is called a Number.

No, I would not say that there exists something in common between those. Cardinality is an emergent property of a set, but that doesn't mean that cardinality exists on its own.

Did Plato ever make the argument that because his Shapes have corners, edges and angles, that therefore corners, edges and angles exist in the same way as the Shapes? I don't recall if he ever described Shapes in those terms.

What does it mean for something to exist then? Does an apple exist in the same sense that a number exists?

There is something in common between two chairs; there is something in common between a red ball and a red book; there is something in common between 21 sheep and 21 dry peas. It is _emphatically_ unclear that each of these three statements of 'existence' have anything at all to do with each other except in the most intuitive sense - that they refer to things which are as opposed to are not. At some point the bar for existence becomes so low that Harry Potter exists - really exists - because we can conceive of the idea of him. The question of whether numbers 'exist' is more than that.

Language, having evolved over tens of thousands of years of the human practical use, is often smarter than us, and where we are forced to say that something exists, it usually does. It means that more often than not something that we describe in words is not merely an idea but has an objective meaning, i.e. has to do with reality and therefore exists, independently from from our minds. Thus not only “objects” exist, but also their properties as well as relations between them. A tip of the needle enjoys the same objective existence as does the needle itself, and this fact is easy to verify by sticking it in you finger. What other existence do you want?

> We can all agree that ghosts don't exist.

I don't agree that the nonexistence of ghosts is definitively settled, and I doubt I'm alone in that here.

For the article to say that the "Existence" claim is "tolerably clear" is rather disingenuous. The basis for this statement in the article is that you can formalize the "Existence" claim with an existential quantifier; but the whole point of an existential quantifier in mathematics is that it has no meaning unless you already know what domain is being quantified over. And in the absence of some well-defined mathematical domain, the existential quantifier basically becomes a tautology: it means "object x exists in the domain of whatever objects exist".

Given this, it's no wonder that such "existence" claims have always been problematic and have always been viewed with skepticism by many non-philosophers and even many philosophers.

Yeah, I think that the reality of mathematical "objects" is very relative. That you can apparently construct an approximation to a Turing machine to any degree says that things behaving like finite mathematical objects can be readily constructed. But I don't think that would satisfy a Platonist.

Your observation here really, I think, solves your own problem.

We shouldn't think of there being a universal quantifier (ie., a universal "Exists") -- but a variety of ones depending on the domain.

We can construct these domains by enumerating objects which share (eg.,) a property. So there is the physical domain of spatio-temporal objects, fictional domain of fictional objects, etc.

The question of whether something exists then becomes a "thin" one: does the relevant quantifier apply? In this sense most things exist -- and this to me feels kinda obviously correct.

Debates are mostly confusions over the domain: are mathematical objects in the physical domain? NO. And no one thinks they are. '2' has no spatio-temporal properties.

If we just drop the idea of a privileged quantifier, everything becomes much clearer.

There is no "really real", or "really exists!". There are just enumerable domains of objects of which we can form claims.

> Your observation here really, I think, solves your own problem.

It's not my problem; I am not a mathematical Platonist, so I have no issue at all with the idea that any claim about mathematical objects "existing" is simply in a different domain from claims about physical objects "existing".

However, for someone who is a mathematical Platonist, there is a problem, because mathematical Platonism appears to be claiming that "exist" is not domain-specific. For example, the article under discussion formalizes the "existence" claim as "Ex Mx" (I don't have the backwards "E" on my keyboard), where "Mx" is the predicate "x is a mathematical object"--i.e., it separates the "existence" claim (the quantifier) from the "domain" claim (the predicate). This only makes sense if the quantifier is not domain-specific but general--i.e., if the quantifier is the same one for all possible domains, so that the existence claim is an absolute one. Which, as I noted in my previous comment, and as you appear to agree, is simply not how existential quantifiers in mathematics are supposed to work.

Maybe the issue isnt the platonists but the physicalists. I suspect platonists would be happy with their own domain.

The issue is physicalists want a privileged quantifier E_physical, and they want a hierarchy of quantifiers where everything reduces to E_physical. Ie., E_mathematical is "really just" a subset of E_physical.

The worry of the physicalist is that Ex means x has spatiotemporal properties; so '2' has to "be somewhere".

Gregory Chaitin has an interesting take on this:

"The only place where medieval theology survives is pure math."

https://www.youtube.com/watch?v=1RLdSvQ-OF0

(The whole interview is actually quite nuanced on the question of whether math is discovered or invented and well worth a watch. There are also interviews with other mathematicians and physicists in the series, for example with Roger Penrose, but most other researchers seem to hold a rather dogmatic and platonist view, which is why I found Chaitin's perspective refreshing.)

Is this at all related to the mathematical universe hypothesis that says the universe is mathematics and that everything that actually is exists is only a mathematical structure?

No, the debate between Platonism and nominalism has been going on for nearly two thousand years (really getting going by the time of Occam.) Max Tegmark's mathematical universe stuff originated with him.

The question here is whether mathematical objects exist independent of our minds. It's the whole "Is math discovered or invented?" question that you see in pop. sci. documentaries. You may find this[1] trichotomy more useful for a quick overview even though that article is about a different topic.

1. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m...

Tegmark's mathematical universe hypothesis is like Platonism taken to the extreme: only mathematical objects exist.

Standard "vanilla" Platonism says: there is an abstract world of pure forms (the mathematical side) and it is imperfectly projected [or other similar kinds of words] down to the reality we experience.

Tegmark asks: why distinguish the two?

The gap between mathematics and physics is big, starting with the digital paradigm that dominates math, but also stuff like relativity and time don't properly exist in mathematics and mathematics routinely forgets about them. Time, law of nature, matter itself - anything disappears when you don't think about it, as if it's solipsism. But it never happens in physics.

Happy to see analytic philosophy trending on HN

What definition of "existence" are they using in this article?

If you like such ideas, the book Reason and Persons by Derek Parfit has some mindbending chapters on whether persons exist separately from the meat, as Cartesian Egos or similar.

I am hoping modern intuitionism and multi-foundational math (think Andrej Bauer) can finally mend the platonism aristotlism divide.

Existence is variative homeostasis.

They do not 'exist' unless they are put in different contexts which affect their properties.

That's... one definition of "exist". There are others.

As all such questions, it depends on what your definition of "exist" is.

What's the other one?

Let's imagine there's a natural number exists between 7 and 8, and it is particularly sneaky so that it won't show up in any arithmetic operations performed by human and their devices. How we can argue it does not exist?

I think you're thinking of bleem, the secret integer between 3 and 4 [ according to the short story "The Secret Number" by Igor Teper https://eyeofmidas.com/scifi/Teper_TheSecretNumber.pdf ].

"Exist as an idea" and "exist as a physical entity", to name two off the top of my head. There are probably more.

Bleem (thank you!) definitely exists as an idea and definitely does not exist as a physical entity. But does it exist as a mathematical object?

No - I checked.

Not yet. (The complex numbers did not exist as mathemadical objects until relatively recently.)

While "checking" it you have interacted with it, so it seems you have confirmed its existence

By the way, many things that "exist as an idea" do exist within the nature itself, like the idea of gravity obviously affects the direction of the water flow in a waterfall.

>Let's imagine there's a natural number exists between 7 and 8

8 is by definition the label for the number after 7. See https://en.wikipedia.org/wiki/Peano_axioms

So maybe you are thinking that there exists very different Set for natural numbers that has this nice properties, /

I think you can try an exercise to define such axioms and rules, see if actually they are consistent and not trivial and then see if they are different then the natural numbers.

But is being variative necessary in order to exist? (In the context of mathematics, for example, the homeo-something seems to be all that's required.)

Numberphile did a nice episode about this topic a while back: https://www.youtube.com/watch?v=1EGDCh75SpQ

I don't think there is a real correspondence between math and some kind of 'mathematical reality' or Platonic heaven. Mathematical notions of truth are just mathematical. If you say something like "2+2=4 is true", you are making the purely mathematical statement "2+2=4", nothing more and nothing less. It's like saying "Sherlock Holmes lives at 221B Baker Street".

It’s more complicated than that.

For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal.

Even if there are two or more of something (not a settled question in physics), the idea that you can add two such groups of two and have it be equivalent to some other group of four of that thing is only an abstraction in our heads.

In reality, if you have two apples in two pockets and I have four in mine, all we can say is that there are eight regions of the universe we like to call “apples”. 2+2=4 isn’t true in the same way that Sherlock Holmes’s address is because the former depends on a shared illusion/set of abstractions.

> regions in the universe

This has nothing to do with natural numbers. What does have to do with them is simple practical considerations like that you need two apples to treat two horses. This is why (small) natural numbers do, in fact, exist - in a very clear practical sense. And if you only have one apple and get bitten by the second horse who is now upset, you can only blame yourself for ignoring such an obvious fact.

At what point does one apple and another one apple become two apples? Is a matter of distance? Likeness? Practicality doesn’t say there must be two apples, just one apple and another one apple. Two (apples) is just an abstraction in my mind.

It's a matter of language and logic. "another" is a contraction of "one other", and "other" means "distinct".

So, given you have "one (element) and one different (element)", at that point, you have, ipso facto, 2 elements.

It's not ipso facto 2 elements. You need to define what it means for there to be 2 elements, or 2 apples. Which goes back to question: "At what point does one apple and another one apple become two apples?"

I can simply say there's only one's of apples. The existence of the number two is optional in this case. Its existence is completely up to my own mind (Distance? Likeness? Etc.). Prove to me in this case that the number two exists outside our own minds.

Natural numbers are objective, observable relations between collections of things (and also between a collection and its sub-collections). They can only be understood by observing differences and similarities between collections. (Similar to how the color red would not exist if everything was red.)

You are right in that one apple and another apple do not make two apples. You need a matching pair of happy horses for the number two to emerge.

>At what point does one apple and another one apple become two apples? Is a matter of distance? Likeness?

Perhaps I am misinterpreting, but it seems to me like this would occur whenever you formed a coherent question. "How many apples are there?" would be insoluble, whilst "how many apples are there in this room?" or "how many red apples are there resting on that table?" would result in an answer belonging to the set of naturals.

Totally agree with you that it’s practical to think of apples and horses as more or less alike. We evolved this mental tendency because it works.

But it is a style of thinking that is adopted because it is useful to human lives on human timescales, not because it is true.

The truth is there are no apples, no people, no objects even. There is just an immense swarm of subatomic particles interacting woth one another across the universe.

If you could observe the particles in that “apple” over a billion years instead of a more human timescale, it won’t make any sense to think of it as an object anymore. The particles started out in stars far and wide, are briefly frozen together in a fruit on earth, and will soon be spread apart widely again.

Some of the particles in the universal soup have combined into a chemical reaction that thinks it is you, and has evolved useful ways of perceiving patterns in other particles (for instance, object permanence). Because this phenomenon (you) is able to perpetuate itself by thinking in this way (perhaps by, among other things, finding and eating apples?), these thought patterns are sticky. Or in your words, practical. But not the whole picture.

This looks like a classical case of someone missing the forest for the trees.

Split one apple into two halves and feed two horses with that.

You are unsure whether "2 + 2 = 4" is real, but certain that "2 + 2 + 4 = 8" is.

For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal.

I'm not convinced. You are defining the number 2 as a material cardinal, and then coupling the existence of the number 2 to the existence of two completely identical material objects. But why would the existence of those objects give rise to the existence of the number 2?

Can't I similarly define that the number 2 is a materialized ordinal, and simply count the revolutions of a moving object, and state that the number 2 is instantiated by observing the periodicity of a single material object?

No, I don't think there is any such implication. Mathematics works by defining some axioms and some rules on those axioms and then people think hard to see what theorems are true in this system. Being able to define something doesn't seem to imply that such a something must exist.

I don't care whether it exists, but I care whether it helps me to reason about things I care about. What changes about life, depending on whether it exists or not? If nothing changes, what's it matter?

There's probably a philosophical term for this attitude. That doesn't really answer my questions, though.

> There's probably a philosophical term for this attitude.

It's usually referred to as quietism[1]. Quietism on this particular topic (and it's generalization) is probably the most popular kind of quietism actually.

1. https://en.wikipedia.org/wiki/Quietism_(philosophy)

> What changes about life

If only that you felt like you have to spend some of your time and energy on writing your post.

We are all back into star dust in 100 years, what’s it matter :)

for me maths is just measurement, including of itself

Well, measurement is physics; mathematics is calculation (in a rather general sense). Incidentally, I think the word "calculus" happens to be more reflective of the nature of the subject than the word "mathematics" which seems to hide a simple notion behind an aura of mystery. "Calculus of Infinitesimals," methods of doing calculations using the notion of the infinitely small, is easier to guess what it is all about than when you first hear the mysterious "Mathematical Analysis."

i think measurement is more general than calculation. when i say mathematics also measures itself, i mean this in a pure-maths sense

I'm getting quite frustrated with this kind of ontological debate. Exist or not exist, why does it matter? On top of that are so many subtle distinctions that the statement "There are mathematical objects." might or might not mean "Mathematical objects exist" – depending of the frame of reference (Meinong vs Quine vs So Many Options) – but no matter the definition it changes nothing. We are still ontologically committed to mathematical objects and treating them as truly real or abstraction which is ontology dependent on other stuff don't really make us use math differently. So what is the point of such debate?

i mostly agree with this but I think the philosophical position might have an impact on how open one is to new methods in mathematics.

If one holds the view that mathematical objects can be understood as 'language games' rather than real ontological objects I think it creates a more pragmatic view on what is permissible, what constitutes a proof, whereas I think the more Platonic views might make someone more purist in how they approach maths.

> Exist or not exist, why does it matter?

Yes, that's exactly how pragmatists[1] answer the question. It depends entirely on what you mean by exists and what you mean by exists always depends on what you're trying to do, i.e. your goal. That's what the whole squirrel* anectdote that William James came up with was about.

1. https://plato.stanford.edu/entries/pragmatism

* The squirrel thing is in section 1 of the linked article.

I've generally found Pragmatism to be pretty parochial and limited, I mean, looking at general statements like this one:

> that a claim is true if and only if it is useful

one cannot be judged for thinking "what good is there for any quest for true/false if all that matters is the usefulness of the thing/action itself?".

I find pragmatism to be too anchored in this socio-material reality of ours, one in which things/actions have to be "useful" (the fact that many of the pragmatists come from the Anglo world, so obsessed with usefulness and utility generally speaking, also helps accentuate this).

Back to the post per se, I think "the exist or not exist" issue/problem when it comes to mathematics is a subconscious sociologic-related plot (I'm only half-joking) of the people studying maths and believing in the power of maths so that they would impose on the rest of us, the non-maths people, the statement which says something like "what we, mathematicians, are studying does indeed exist and stands at the roots of the (scientific) world as we know it! Give us symbolic power in return!".

Tell some lay-person that what a mathematician studies is in effect just a figment of his/her imagination, and one of the questions posed in return by that lay-person might be: "why should we believe what mathematicians say more than what poets say?".

It sounds like you could have a narrower idea of what utility is than I do. For me, this is really about definitions relative to systems and those systems serving out goals. That's where utility comes in: Whether something is true or not depends on the system in which we judge it (i.e. relativism) and what system we use will depend on our goal. Our goals obviously depend on what we like and want, so axiology is clearly the root here.

You'll run into this immediately when you study logics and find out that different ones can prove different theorems thus they have varying use cases, similar to the way programming languages do. If you want to predict the future, science is the tool you want but if you want to court someone then maybe poetry is your tool.

It's not that anything has to be useful, it's just that analyzing concepts in terms of utility can clarify what we mean and illuminate the criteria necessary to decide certain questions.

I agree with your second-to-last paragraph. Everyone wants what they study to be proven to be "real." It protects the prestige of the field. I always figure there's a conflict of interest going on when you find most mathematicians are Platonists but this unanimity doesn't carry over into philosophy where careers and social lives don't ride on that result [1]. I hope my previous paragraphs clarified when I think the layperson should trust the mathematician.

1. https://philpapers.org/surveys/results.pl

> If you want to predict the future, science is the tool you want but if you want to court someone then maybe poetry is your tool.

I agree and I definitely see your point, what I'm saying (or what I'm trying to say, at least, English not being my mother-tongue and me not being a mathematician nor a philosopher) is that going one (or two steps) back one may ask him/herself about the very concept/idea of utility, more exactly on what "premise" (so to speak) we assign "goals" to the "reality" surrounding us, why do we assume there is an "utility", why do we assume there should be "goals"? Why should we instrumentalize said "reality" by seeing it through the prism of "utility" and "goals"?

I know that taking that many steps back from "natural" concepts like "utility" and achieving "goals" risks turning everything into empty metaphysics and, worse (from the pov of a mathematician or of a scientist more generally speaking) into mysticism, but imo at the higher level the mysticism of the neoplatonists (i.e. Plato's ideas taken several steps further with some pre-socratic elements added in) plus the ideas of some pre-socratics themselves are more closer to the "reality" of it all than what we ended up adopting (basically Aristotle whom we refined/corrected in some specific points). Basically I think a guy like Heraclitus was closer to grasping the "reality" around us than Aristotle ever was, but had we gone the way of Heraclitus we wouldn't have had computers and rockets and all, computers and rockets that were made possible by Aristotle's logic (and by the way how he perceived the logic process per se).

Again, imo, Plato stands between the pre-socratics and Aristotle, but because we chose Aristotle's way and we basically built our world on Aristotle's way of thinking when we read Plato we tend to focus only on the part of his philosophy that in a way confirms this "world" that we built, so to speak (with this correlation between Plato's ideas and mathematics being just an example for that).

Anyway, these are just some ramblings :)

> more exactly on what "premise" (so to speak) we assign "goals" to the "reality" surrounding us, why do we assume there is an "utility", why do we assume there should be "goals"? Why should we instrumentalize said "reality" by seeing it through the prism of "utility" and "goals"?

We don't assume. It's just an empirical fact of our life that we have desires and goals. Don't you have desires and goals? I do.

> Basically I think a guy like Heraclitus was closer to grasping the "reality" around us than Aristotle ever was, but had we gone the way of Heraclitus we wouldn't have had computers and rockets and all, computers and rockets

This is an interesting idea. It reminds me of the Two-Truths Doctrine of Buddhism as interpreted by the Madhyamaka school. They believed that there was a provisional truth useful for doing doing daily stuff (like how you see Aristotle) and then an ultimate truth (like how you see the pre-Socatics.) I think there are monists in Western traditions who have similar views but I couldn't name any.

During my uni days I had a math lecturer who stated that it was a matter of great pride to him that all the math he created was pure and had absolutely no practical applications.

With this in mind what does usefulness has to do with truth?

Alternatively, his results where doubtlessly useful to other mathematitians in their research, does that make them more true?

> I'm getting quite frustrated with this kind of ontological debate. Exist or not exist, why does it matter?

One's philosophical / metaphysical worldview determines how one views reality.

As an example, some variations of Islam follow(ed) occasionalism:

> Occasionalism is a philosophical doctrine about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God. […] The doctrine states that the illusion of efficient causation between mundane events arises out of God's causing of one event after another. However, there is no necessary connection between the two: it is not that the first event causes God to cause the second event: rather, God first causes one and then causes the other.

* https://en.wikipedia.org/wiki/Occasionalism

Whereas Christianity rejected it and went with secondary causation:

> Secondary causation[1][2][3] is the philosophical proposition that all material and corporeal objects, having been created by God with their own intrinsic potentialities, are subsequently empowered to evolve independently in accordance with natural law.

* https://en.wikipedia.org/wiki/Secondary_causation

So in in the first case asking "Why did X happen?" you answer "God willed it.", while the second case you would say "There is something in Object A that interacted with Object B." The latter then leads you down the path of examining objects and their relationships, as opposed to chalking events to spirits, gods, or God exclusively. Plants/crops growth because of something with-in themselves and not because of Ceres / Demeter willed it.

Without this worldview, you don't operate under (e.g.) the zeitgeist of being able to investigate Nature:

> That this objective reality is governed by natural laws;[35][36]

> That reality can be discovered by means of systematic observation and experimentation.[35][36]

> That Nature has uniformity of laws and most if not all things in nature must have at least a natural cause.[36]

* https://en.wikipedia.org/wiki/Naturalism_(philosophy)#Provid...

So on a day-to-day basis you may not deal with these axioms/beliefs, but they are there nonetheless.

"One's philosophical / metaphysical worldview determines how one views reality." I fully agree but I fail to see how it is true in the case of ontology of mathematics. All ontological frameworks will have to explain the existence of the same entities so what is left is the difference of vocabulary and maybe cultural practices embedded in each view, but the reality will be the same. You won't meaningfully deny the existence of triangle only because you changed your ontology.

No, but it might change your perspective on things and lead you to explore a different set of ideas (something that may end up being quite important when doing scientific research?).

Lots of folks are asking questions in that vein:

* https://aeon.co/essays/does-philosophy-still-need-mathematic...

Do imaginary numbers exist? They used to be seen as "as fictitious or useless" until Euler came along:

* https://en.wikipedia.org/wiki/Imaginary_number

What about octonions in the present day?

* https://news.ycombinator.com/item?id=17575585

* https://en.wikipedia.org/wiki/Octonion

It's kind of hard to tell what idea will lead where.

> don't really make us use math differently

Even is we suppose what you're saying is correct — the goal isn't really to change how we use mathematics, but to understand what math is in a deeper explanatory sense. If that's not a project that interests you, that's totally fine.

It would be odd if we made deep advances in understanding math at an explanatory level without having some kind of translation to how people do math.

Maybe, but it doesn’t seem like a requirement. Like having an understanding of jazz at the level of music theory vs being a jazz musician.

It's also difficult for me to believe that deep advances in music understanding would fail to translate to actual music.

Could you give me an example of meaningful insight of what math is that relies on such ontological discussions? Since my issue is not with ontological debates as such but rather ontological questions formulated in a manner which is impossible to meaningfully answer. I think we can learn great things about mathematics (and world) by looking how people do math, or how people speak about math, but asking what math is seems to me to be a way of framing that instead of providing insights is dividing people into camps based on aesthetic preferences.

We often investigate things without knowing what value, if any, doing so will produce. It seems odd to require a business plan up front here. People do have them, entire metaphysical projects rest on this question and they can definitely tell you why it matters to them, but I question whether they should have to.

I mean philosophy, metaphysics and philosophy of mathematics have over 2k years of history and the only example of the influence that I know it had was that ancient Greeks didn't want to use 0 in accordance with the rule that being is and non-being is not, similarly with Pythagoreans hiding the existence of irrational numbers. If you know of some examples which had positive effect then I would really like to know about them and this is not a sarcasm or anything. I just fail to see the importance of the debate.

Search "Brouwer" in this thread.

Yea I saw that, I will check him out, thanks.

> So what is the point of such debate?

This is an incredibly parochial point of view.

Kant's seminal "existence is not a predicate" argument has had a foundational impact after he put the rationalist vs. empiricist debate to rest. In fact, it's extremely deep, arguing that the P in ∃xPx can't be "exists." Modal logicians have tried to come up with more clever ways of circumventing this, e.g. E(t) := ∃x(x=t)—although this isn't entirely non-problematic, either.

For that matter, the question of anything existing is profoundly human, so asking "what is the point" kind of misses the point.

"This is an incredibly parochial point of view.", why? I don't really see how what you said and what I said connect.

If one thinks that certain mathematical are mere constructions, one might also reject certain mathematical methods. Take intuitionism as a case: Brouwer thought that mathematics was the product of the human mind, and restricted maths to whatever is constructible in a very strict sense. Finitism adds even further constraints, disallowing infinitary methods altogether. In these cases, the position seems to be that we cannot hold on to the usual commitment to mathematical entities.

> Brouwer

I want to also point out that his philosophical attitude turned out to also be very productive in the end and lead to significant contributions in our understanding of type theory which are applicable today in theorem provers and functional programming languages. He also gave new proofs of various theorems independent of various other results and made room for a lot of advances in logic. So, I think even the side-effects of pursuing this kind of project can be rather valuable and have obvious practical impact.

I would also characterize myself as more of a pragmatist, and to me, the pragmatic version of that debate is around a reframing of the question, which becomes the classic: "Why is mathematics so (unreasonably) useful?" I.e. why does math work at all?

Certainly looking at what mathematics is or could be in relation to the rest of the universe looks like a legitimate research area to then potentially guide the evolution of our mathematical tools.

I'm particularly interested in how one can reconcile naturalism/physicalism/materialism/monism with non-platonism/nominalism, i.e. if there is no such thing as Platonic ideals, what's the "physical" nature of math?

(Using lots of quotes here because I'm being lazy and not super careful with the terms I'm using — I'm aware.)

Why does anything matter

The correct answer is: "why give a fuck?".

The only practical value of having an answer to this question is giving philosophy students something to write about.

Mathematics is a human activity, so whether you think it's "exploring" or "inventing" mathematics, you would agree that we aren't pursuing the exploration or invention in all possible directions.

So let's look at what we, the humans, actually do.

Any mathematical object ever defined is ultimately an abstraction and a model of something the humans actually experience in the physical world.

Number theory is abstract, numbers are not. You can count your sheep in a pasture; you can count your neighbor's. Natural numbers model the idea that neither of you has more than the other.

This is just a model, of course; you have a sheep named Dolly which is dear to you, and your neighbor has one named Sue which gives more wool than any sheep in the county.

So you really don't have the same thing, because these sheep are not interchangeable. But we have a concept — natural numbers — which models the idea of sameness if we, for a second, forget that.

It's an imperfect model; the concept of something that can grow without bound is an outcome of this imperfection. When you use numbers, you might as well assume that 10^(10^100) is the largest number there is. Numbers of that scale are so large, that all computers in the world taken together aren't enough to just store this number. But modeling the concept of "not knowing what the bound is" is modeling an experience we run into all the time. 640KB isn't enough for everyone.

We dispense with this abstraction with ease, though. Most calculations in practice are done with 64-bit "numbers". These numbers don't behave like the one in math do, but it's OK.

Even simpler, our clocks have just 12 numbers, even though the time doesn't stop running.

Euclidean geometry can be verified experimentally. Cut out a triangle out of paper; tear off the angles; put them together, and see that they make up a straight line.

It's an imperfect model, and you don't need Einstein or Lobachevsky to tell you that. Long-distance sailing made people run into this model breaking. The corners of a triangle you cut out of orange peel with a straight knife won't make a straight line.

But that only gives rise to other kinds of geometries being interesting to us.

People do math because it's beautiful, and something that doesn't connect to any human experience is decidedly not beautiful.

Take a deck playing cards, or just a dozen cards, and come up with a set of rules of how a pair of cards can be traded for a single one. You can make different rules depending on which of the two cards is on top; 2-for-1 is the only restriction.

We could come up with an immense number of complicated systems that way. But most of them would be so arbitrary, they won't connect to anything. So nobody cares about systems like that. Unless they come from something else and therefore have more than an arbitrary structure.

Like finite symmetry groups, e.g. rotations of a cube. You can model that with such card rules. Then it's interesting.

So the question of whether mathematical objects exist is beside the point.

The reality is that all mathematical objects that humans care to study are, ultimately, modeling an experience that does.

And this is consistent with mathematics truly being an art, not a science.

Because art that doesn't abstract human experience in some way is not something we care about. Strings of letters spit out by a random number generator aren't an exciting read.

That's what Vladimir Arnold means when he says, seemingly controversially, that mathematics is a branch of physics (in which experiments are cheap)[1].

It's that mathematics that doesn't connect to the world we live in is too boring to exist - or be invented and remembered.

[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

> People do math because it's beautiful

That's pretty much the only thing I disagree with* - people do math because you often can't build a freaking bridge without it.

*The rest is not even wrong.

The entire point of what I was saying is that math that doesn't connect to reality isn't beautiful.

There's beauty in figuring out the exact shape a hanging rope takes just by thinking about it.

When I say "do" math, I mean "come up with new math".

People use math (..and fund math) because you can't build bridges without it, but people do math because it's beautiful.

>* The rest is not even wrong.

Would you explain this part of your comment?

I write this as a mathematician (math PhD, publications, yadda yadda), and what I wrote above is my attempt to translate Vladimir Arnold's article[1] into laymen terms (my non-math friends found that article inaccessible).

Vladimir Arnold is the mathematician (immense contributions to the field that spawned entire branches of math).

All the statements concerning math concepts (numbers, card rules, etc) above are correct - and easily falsifiable; so "not even wrong" hardly applies there (unless you meant it to say that it's not wrong, which has a different meaning).

[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

Much of what you were saying was hard to agree or disagree with. BTW I find Arnold's piece easy to understand (and appreciate) even for a (somewhat educated) layman.

Mathematical platonism is ridiculous quasi-religious mumbo-jumbo that keeps philosophers employed.

You've only ever explained a circle to other humans. You've only ever learned about circles from other humans, or artifacts (not circles!) authored by humans.

No human has seen a perfect mathematical circle with their senses, because they don't exist as physical objects outside of our skulls. They exist as physical objects inside our skulls in the form of neural correlates and latent representations.

A circle is like a meme. It's compelling because it's so simple (to us anyway). But, try to really explain a mathematical circle to a six year old and especially how it is different from the representation of a circle that you've just drawn. Watch even a smart kid struggle a bit with this, before getting it. It's not as simple as we might think, having mastered the concept. That doesn't mean it has an independent existence in some realm that is not embodied. It just means the concept is hard, then simple due to its abstractness and otherworldly simplicity / power balance.

EDIT> apostrophe

Sure but your argument is so broad that it applies to everything then. Apples don't exist, cars don't exist, the color blue doesn't exist. By the same standard that you hold circle to, you have never actually seen an actual apple, or blue, or a car.

If you hold any of these extreme views you end up with quasi-religious mumbo-jumbo. The challenge is at what point do you draw the line so that one can say that an apple really does actually exist, but the square root of -1 does not exist? What about an electron or a gluon, do those actually exist in the same way an apple possibly exists? What about a virtual particle? Do those actually exist or are they also memes whose only existence is as neural correlates? If their only existence is as neural correlates, does that mean that they can't exist as "transistor" correlates? What about other forms of matter, can they also correlate so as to give an existence to an apple or is there something special and unique about neurons?

It's the investigation of that question itself that is of interest, and I think quite valuable, both in principle as well as useful from a programming point of view.

> It's the investigation of that question itself that is of interest, and I think quite valuable, both in principle as well as useful from a programming point of view.

I agree, but I believe our progress in this area is hindered by unjustified ideas like platonism inherited from our history of religion and spiritual belief. They bias us towards unnecessary non-materialism.

> Sure but your argument is so broad that it applies to everything then.

No. When I said no one has really seen a circle, I wasn't distinguishing between sense impressions / predictions and some external reality. As you point out, that would be problematic. But there is no true apple form. Granted, there may be stereotypical apples.

A mathematical circle is different. It has equal radius everywhere in a continuum (real numbers). It has no thickness. These are essential features of a mathematical circle. I have never seen such an object. Neither has anyone else. It can't be built out of matter, which is probably the appeal of platonism. It does exist in material form in our heads, not outside. It can be reasoned about. We can draw approximations to it. This changes nothing for working with math, as others have pointed out. We should obviously choose to speak in terms of circles when doing math, instead of specifying the exact neural correlates of circle, but it's a matter of abstraction and utility.

Regarding fundamental? particles, we recognize that these are features of our current model of reality.

We can certainly talk about abstract concepts, but there is no reason to think that they exist without physical correlates. That doesn't lessen the usefulness of the (possibly leaky) abstractions.

Regarding other substrates. Yes, figuring out what other substrates could possibly support what I experience as consciousness, and what others report as consciousness is very interesting/useful. I'm working on this now.

I think we get in trouble though when we define consciousness too broadly.

I think that your insistence that your own existence is substantially different than that of an ideal circle is the real "quasi-religious mumbo-jumbo".

"Physical reality" is just your way brain's way of interpreting the mathematical rules that constitute our existence. Yes, there is an external reality that can bite, because it is composed of rules that apply equally to all the minds within it, but it is the minds that make a "reality" rather than simply existence. A circle has no mind, no viewpoint to make its existence feel real, but it exists no less.

> "Physical reality" is just your way brain's way of interpreting the mathematical rules that constitute our existence

You state this as though it is a fact, but really this is the main point of disagreement. You cannot take this as given.

But isn't philosophy (and religion) nothing but disagreement? (If philosophers agreed with each other, we'd very few philosophy books; in contrast, despite the fact that there are many, many mathematics books, those are pretty much all saying the same things.)

Heh... and neither can you take it as a given that physical objects outside of your skull have a more substantial existence than mathematical ones.

Let's do a simple thought experiment... Do you believe that GAI is possible on conventional computers? If you do, then you must believe that a mind not significantly different from yours can exist inside a particular configuration of Conway's Game of Life (since Turing machines exist in GoL). But GoL is purely mathematical, so the mind-in-GoL is composed of, and senses and interacts with, objects "external" to it which are purely mathematical.

But... 1) such objects would seem "physical" to the mind-in-GoL, and 2) both mind-in-GoL and its entire reality are fully deterministic and their existence is thus entirely independent of them ever being "simulated" (or instantiated) on an "actual" computing device. If they can exist, they can be said to actually exist, regardless of any external (to them) observer observing them. One of us running such a GoL configuration on one of our computers is just a way for us to create a window into that existence, not to "bring it into being".

In short, if you believe that GAI is possible and that your physical reality is posited within an essentially deterministic Universe (and here it does not matter much how you resolve the quantum dilemma), then there is no reason to give physical objects a privileged position over mathematical ones. In other words, your position is based on nothing but "quasi-religious mumbo-jumbo", an unjustifiable faith that your existence is special.

> But GoL is purely mathematical

Is it though? Humans play GoL on computers made out of matter, think about GoL using brains made out of matter, and learn or communicate about GoL using artifacts made of matter. We can port GoL to various devices, so it seems that the essentials of GoL are not tied to a particular physical substrate. But then again, these are only approximations to GoL in that storage, etc. is limited. Anytime we actually run GoL, we're doing it in some embodied form. GoL doesn't play itself in some non-material plane.

I think we get in trouble with these sweeping thought experiments and extrapolations. I might agree in principle that a mind like mine could exist in GoL, but it could turn out that to implement it would take some enormous combinatorial number multiple of universes. Then does it exist in a form that could operate, or only in a form that I can sketch out to others without understanding or giving any details?

We really need to be careful about 1, 2, ..., infinity style arguments. They're second-nature to mathematicians, but the history of mathematics is full of cases where we had to clean up the mess left over after inviting infinity to the party.

EDIT> "2) both mind-in-GoL and its entire reality are fully deterministic and their existence is thus entirely independent of them ever being "simulated" (or instantiated) on an "actual" computing device. If they can exist, they can be said to actually exist"

How is this not just restating the same unsupported argument? Now we'll just have to debate what "exist" means, which is more or less where we started.

EDIT 2> This is what I mean by keeping philosophers employed. Instead of making progress and deepening our understanding, we just end up debating and then agreeing-to-disagree on what word X means, and what it means to mean, and whether meaning can be said to exist, and ..., and ..., recursion

Please make your substantive points without name-calling.

https://news.ycombinator.com/newsguidelines.html

One could say that mathematics does not study (abstract) objects; rather, it studies relations that do, in fact, exist between (real) objects. Even a circle is merely a relation - that between what lies on the circumference and, for example, what's at the center (or the axis, if we speak about a cylinder); this also manifests itself in the form of rotation, ostensibly a physical phenomenon.

That's kind of a cop out, though. Relations are presumably abstract objects, too?

Relations often realize themselves as physical interactions.

They sure do, much like words correspond to physical things, but words are still "abstract objects" (if you subscribe to Platonism, that is).

But relations are physical; the fact that they are the sole focus of mathematics (unlike physics where focus is also on objects and processes) does not change that.

In what sense are they physical? Can you point to one?

In the sense that they exist in nature. Natural numbers (correctly understood as relations) exist, for example, as atomic valences which permit certain interactions between atoms while preventing others. (This has nothing to do with the Platonism BTW.)

If to you relations exist in nature therefore they are physical, there is a physicalist assumption in there which doesn't go without saying.

Now, I happen to have sympathy for that position, but it's hardly as straightforward as you seem to imply.

Where it does have a lot to do with Platonism is that positing that relations are not abstract things but physical entities presumably entails that you can show what a relation physically is. And an instance of a relation isn't the relation. As soon as you go into "for example", you're not talking about the object itself anymore.

(To better see that relationship, see: https://plato.stanford.edu/entries/abstract-objects/)

So what kind of physical object is a relation?

It pays to draw a distinction between objects and relations. Relations manifest themselves when more than one object is involved. A pair of objects is an object; it is a physical relation between pairs, the isomorphism if you will, that we call "number 2"; it manifests itself in cases when pairs can physically "snap" together solely due to the fact that their cardinalities match. I think it is the fact that a number (a count, in this case) is not tied to a single object (e.g. a pair) but rather expresses the potential of a certain physical interaction between several objects that makes understanding this difficult.

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