20 ms·

Does mathematics need a philosophy?

102 points by furcyd 4y ago

damiankennedy 4y ago
I don't think any philosophy is comprehensive enough to include all of mathematics and any partially complete choice is going to be subjective. You could even replace math with "x" and come to the same conclusion.
bugbuddy 4y ago
Betteridge's law of headlines dictates that the answer must be no .
- pixl97 4y ago
  Hence proving Betteridge's law is not universal.
monero-xmr 4y ago
I can't imagine a universe where 1 + 1 = 2 is incorrect. No matter what the constants are of any universe, in the abstract having 1 of something added to 1 of something is 2 of somethings. That is pure logic. If you can establish the axioms of zero, 1, adding, multiplication, etc., you can then continue into the rest of mathematics and also computation. As a side note I don't believe in infinity as anything other than a mathematical concept nor do I believe in parallel universes. I do believe there is a finite amount of universes (if more than one universe exists) and therefore a finite amount of matter. I think if we found intelligent life, they would have discovered precisely the same mathematical facts. To circle back to the philosophy of mathematics - pure math is pure logic. I think discovering God is more about physics and the Big Bang rather than via mathematics. However mathematical facts can be so pithy and perfect, that sometimes you wonder if God made them, and you are discovering mysteries of the universe as you find more mathematical discoveries. Discussing God on HN is not common, but given the linked post and its subject matter, it's hard to avoid that topic.
- photochemsyn 4y ago
  In symmetry, 1 + 1 = 1, where 1 (or c1) represents a 360 degree rotation around an axis. Also, 2 + 2 = 1, and 4 + 4 + 4 + 4 = 1 (180 degree rotation = 2, 90 degree = 4).
  - monero-xmr 4y ago
    You are representing an abstract concept (circles) using base 360. That’s unrelated to the fundamental concept of adding 1 and 1 to get 2
    - ausbah 4y ago
      fundamental is just whatever axioms you choose, not your gut intuition of what "makes sense"
  - Barrin92 4y ago
    mind you Platonism concerns the existence of mathematical entities, not their referents. If you take "1" to represent a natural number in one case and a rotation in another, that's not what matters. You could use hieroglyphics or emoji if you wanted. The question is, is the mathematical object and relation itself real? Formalists claim that mathematical expressions represent nothing at all, they're just syntax. The platonic question is, is there a mathematical symmetry where the identity operation does not hold? Is there Euclidian mathematics where a 360 degree rotation does not return you to your original state, where one thing is not equal to itself, and so on.
    - DiscourseFan 4y ago
      I mentioned this in another comment, but Platonism only is concerned with how humans are able to communicate with each other using shared concepts, if the "entities" exist as you say, most people who study philosophy would agree that insofar as they exist, they exist in people's heads (since, something existing in your mind is still real in so far as it has a material reality).
hackandthink 4y ago
Mathematics has a philosophical bent. Philosophy tries to make implicit understanding explicit and usually fails in interesting ways. (A part of) Mathematics tries to get the essence of numbers, calculation or the continuum and formulates corresponding axioms. So Mathematics tries to make the implicit understandig of for example natural numbers explicit. Mathematics doesn't fail usually because the scope is much smaller. I think some Mathematicians whose scope is very wide should be regarded as philosophers. Per Martin Löf: "Martin-Löf type theory is a formal logical system and philosophical foundation for constructive mathematics" (1) William Lawvere: his topos theory amalgates Logic and Geometry and leads to a kind of natural philosophy (2) (1) https://plato.stanford.edu/entries/type-theory-intuitionistic/ https://plato.stanford.edu/entries/type-theory-intuitionisti... (2) https://ncatlab.org/nlab/show/William+Lawvere#RelationToPhilosophy https://ncatlab.org/nlab/show/William+Lawvere#RelationToPhil...
- alwayslikethis 4y ago
  > Philosophy tries to make implicit understanding explicit and usually fails in interesting ways. > Mathematics doesn't fail usually because the scope is much smaller. Another way to look at this: we try to make implicit understanding explicit in a large number of fields. The fields in which it succeeds we call mathematics, and the other fields in which it fails we call philosophy.
  - srcreigh 4y ago
    This hits to the core of CS, actually. Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable. Plug for implicit understanding though. Your very claim is not explicable; you can’t explicate the non existence of explicability. You can’t formally denote areas for which there’s no formal descriptions. This is the escape hatch out of logical realism (and yeah, that’s not an explicable claim, either; how fun!)
    - blackbear_ 4y ago
      > Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable. I am not sure that this is the case actually. Many results in mathematics only relate to the (non) existence of certain objects, without explicit recipes for finding (i.e., computing) them. And obviously philosophers started arguing whether an object that cannot be found actually exists or not :)
burna_aws_acct 4y ago
Maybe it's the directionality of the question that is confusing. How about, Does Philosophy needs Mathematics? To which I would answer, yes.
- DiscourseFan 4y ago
  I would disagree, simply because math deals with a fairly restricted sphere of inquiry, whereas philosophy might ask questions about things that mathematical logic, being restricted axiomatically, can't necessarily touch upon. But in truth, we should attempt to be interdisciplinary, I think when people work together on things and try to share their knowledge across field the most progress is made.
- marginalia_nu 4y ago
  This is as old as Plato though. His academy supposedly had a sign saying "let no man ignorant of geometry enter here". But it is very much bidirectional. Philosophy is necessary to understand the foundation on which mathematics sits. If you don't understand concepts like truth and existence, can you really meaningfully engage with work that deals with demonstrating truth and proving existence? The problem of universals is also very relevant as the article suggests. When prodded, you will probably be able to infer answers for what it means to exist, or what makes something true, but those are philosophical standpoints. They have arguments and counter-arguments. They imply an entire world-view. It's fine to make a philosophical standpoint, I think it's more questionable to do so without realizing their context, or even that you're even doing so as is often the case in the natural sciences. If you haven't studied philosophy, you tend to be very blind to the philosophical assumptions you and everyone around you is making. This is especially a problem as STEM education is absolutely steeped in philosophical assumptions and positions without offering the language or tools to recognize and discuss them, so they sort of seep into your mind through some sort of cultural osmosis. I think it's often perceived as though these standpoints (e.g. logical positivism) are the somehow the result of science, or inherent in the scientific process, but that's just not so. They're a means of interpreting the results, one of many.
  - kerkeslager 4y ago
known 4y ago
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DiscourseFan 4y ago
I have a great respect for this author, but I do have to note that the debate of "constructivism" vs. "platonism" doesn't really exist in Philosophy per se, because it relies on a kind of basic misunderstanding of Plato in general. At least since Kant, the general consensus in Philosophy about Plato's "forms" or ideal objects is that they are conceptual objects, famously Kant argued that while reality in-itself is inscrutable, the means of judgement whereby we organize the world into concepts (beginning with the intuitions of time and space) are not, and by analysis of judgement we can begin to better understand the process of reasoning and therefore more rigorously understand the means by which empirical judgements are made (and thus further our understanding of empirical reality). Of course, many have attempted to answer the question of the psychological/biological explanation for how humans have shared concepts like this and shared, intelligible language. The most prominent thinkers of the 20th century, on both "sides" if you will, were Lacan on the "continental" end, and Chomsky on the "analytic" side. (Even though most Anglo-American--i.e. Analytic--philosophers would not accept Chomsky's logic as valid, and Lacan often referenced Anglo-American philosophy in his work...the categories are basically meaningless at this point.) But there are many other names, many people asking the above question, and I will not say who's thought attracts me the most, but I will say that most people I meet who study math are not familiar with them or the intellectual trends they are a part of, and they do not have a rigorous enough understanding of what philosophy is, really, to make claims about the usefulness of philosophy for their work--or even debate it, for that matter.
- techno_tsar 4y ago
  I believe that "platonism" in this article refers to mathematical platonism, i.e. the view that mathematical entities actually exist, as opposed to mathematical formalism, i.e. the view that mathematical entities are made up. https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat... https://iep.utm.edu/mathplat/ https://iep.utm.edu/mathplat/ This view is only called platonism because it parallels Plato's thoughts, but it is not an explicit reference to Plato's original philosophy. Modern platonism (after Kant) exists as an attitude about a kind of non-mental realism of abstract objects, and so it is with mathematical platonism. These days, there are very few Platonists (those who subscribe to the philosophy of Plato), but many platonists. As for the second point: A PhilPapers survey showed that most professional philosophers of mathematics are platonists, and most mathematicians who are familiar with the debate lean towards platonism. It is much more intuitive to understand how mathematical claims are true if you believe that numbers are real objects in the fist place, which in turn allows us to use the same accounts of truth we would normally use for non-abstract objects. At least prima facie, mathematical truths like 2+2=4 do not seem like they are different types of truth than empirical truths like "Biden is currently president". If mathematical objects like 2 and 4 weren't real, we would have to come up with a new account of truth that applies specifically to mathematics to explain what makes its statements true, which while possible, is a lot less elegant. Obviously, that doesn't refute platonism's alternatives, but in the words of a friend, it would really suck if he spent his career studying something that was entirely fictional, so he better be a platonist.
js8 4y ago
I have a good example why practical mathematics needs philosophy. In mathematical analysis and computer science, natural numbers and infinities have different character. In most of mathematics (like number theory or analysis), it's the countable infinite objects and their properties that are being studied, like natural numbers or infinite sequences. The finite sets themselves are considered somewhat trivial (natural) objects, and uncountable sets are mostly a universe where things happen. As a consequence, for mathematicians, all natural numbers are the same, no matter how large. In computer science, I would argue, the view is quite different. The interest of computer science is in medium-sized but finite sets (roughly between 2^6 and 2^(2^6)). The small sets of size less than 2^6 are trivial (you can brute force them without a computer), while with sets that are larger than 2^64 we almost don't need to compute with (they are like limits what can a DB store), and numbers above 2^(2^64) are pretty meaningless. So you can see, different natural numbers matter differently, some are important and some are less, they are not treated as identical objects. Since computer science is kinda part of mathematics, you can see the philosophical focus is different.
- bibanez 4y ago
  Still, what you prove for all naturals in Math still works for a finite number of them so the limitations in CS are just arbitrary, I don't think they matter much philosophically
  - js8 4y ago
    They matter in what problems each field focuses on. And the idea that the natural numbers are all the same, but infinities have different character can be seen as arbitrary as well (cf. set theory).
    - ogennadi 4y ago
      We need INFINITY because no matter how much storage we have, we'll always have problems that need even more storage to solve?
- srcreigh 4y ago
btilly 4y ago
My controversial take. Should mathematics prove to be in need of a philosophical insight, it will not come from a philosopher. The reason is simple. Most fields of study only find themselves in need of philosophical insights when they are internally driven to crisis. But when this happens, it is those who are part of said crisis that have the best insight as to the actual needs revealed, and it is from them that critical ideas come. Even if a philosopher happens to make a good point in advance, the philosopher will be ignored until the field of study has driven itself in crisis. After the crisis has passed, the philosopher's insight may be discovered to be relevant. But it is unlikely to have been noticed at the time. This is not just an abstract theory. It is borne out by history. For example calculus as envisioned by Newton and Leibniz had foundations in handwaving and wishful thinking. Bishop George Berkeley was absolutely right to complain about "ghosts of departed quantities". But it didn't become a crisis until the following decades when Joseph Fourier added together a bunch of sin and cosine terms (all well-behaved functions) and came up with a square wave (emphatically NOT a function by then current standards). While Fourier proved Berkeley right, Berkeley's complaint did nothing to help solve the crisis at hand. Instead we followed a decades long path through Cauchy defining infinitesmals in terms of sequences approaching zero, and then Weierstrass came up with limits when that followed. We dutifully put a passing mention to Berkeley in a lot of textbooks for being so prescient, but he was otherwise irrelevant. And so it proved again at the end of the 1800s. As mathematics again had a foundational crisis we came up with the three basic schools of mathematical philosophy that are still discussed. Platonism is a description of what mathematicians believed by default, though those who officially hold to it tend to be religious like Kurt Gödel. Formalism was essentially created by David Hilbert, a mathematician. Its opponent, Intutionism (which falls under Constructivism), was put forth by L. E. J. Brouwer - another mathematician. And so it has proven in other fields. When physics had a philosophical crisis with interpreting quantum mechanics, most of the important interpretations were produced by physicists. When evolution theory had foundational crises around how to incorporate genetics, most of the critical contributors to The Modern Synthesis were biologists. (Though one of the critical early contributors was R. A. Fisher, a statistician.) But philosophers were conspicuous by their absence. All that said, the people who come up with those key philosophical insights are often personally interested in philosophy. Having a mind that is capable of such philosophizing does seem correlated with curiosity about philosophy. BUT engagement with subject expertise during the period of crisis seems to be the more important criterion for important philosophical contributions.
richardanaya 4y ago
Mathematicians always have a philosophy. The only question is whether they know themselves well enough for it to be explicit.
- n4r9 4y ago
  I'm inclined to agree, and to extend the statement to physicists as well. Sometimes I even wonder if they're being disingenuous, for example Stephen Hawking on the one hand says "Philosophy is dead" and on the other hand actively engages in the debate over philosophical realism: https://medium.com/paul-austin-murphys-essays-on-philosophy/stephen-hawkings-model-dependent-realism-2b2973af41a6 https://medium.com/paul-austin-murphys-essays-on-philosophy/...
  - GoblinSlayer 4y ago
    Maybe he thought philosophy is an attempt to figure out cosmology from pure thought.
- Koshkin 4y ago
  Philosophically speaking, everything is (or can be) a subject of philosophy, including philosophy itself.
deleted 4y ago
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practal 4y ago
I think philosophy is relevant to mathematics in the sense that one sometimes must be willing to step back from what one is doing, and ask deeper questions: Am I making assumptions here that are actually not true? This is a difficult thing, because how can you free yourself from assumptions you don't even know you made? Platonism can help here, because if you are trying to create a thing you know should exist, but you fail to do it, although you try again and again and again, then at some point you can either give up, because you think it is all arbitrary anyway, so why bother? Or you just keep digging, reevaluating your assumptions until something clicks.
guender 4y ago
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wslh 4y ago
But... mathematics has already a philosophy.... see Thomas Tymoczko [1] and traverse the links ;-) A great compilation is "New Directions in the Philosophy of Mathematics" [2] [1] https://en.wikipedia.org/wiki/Thomas_Tymoczko https://en.wikipedia.org/wiki/Thomas_Tymoczko [2] https://www.amazon.com/Directions-Philosophy-Mathematics-Thomas-Tymoczko/dp/0691034982/ https://www.amazon.com/Directions-Philosophy-Mathematics-Tho...
- kerkeslager 4y ago
  "Philosopher says math is philosophy" isn't proof that math is or needs philosophy.
callesgg 4y ago
Math is a form of and a part of philosophy. If you can’t see that, you are the one who needs philosophy.
Pinegulf 4y ago
Isn't mathematics just form philosophy where axioms are accepted?
- nigamanth 4y ago
  There are no axioms in philosophy, in math 1+1 = 2. That's an axiom. Within philosophy however, there's not one type of philosophy that is right, it's subjective so no.
  - msm_ 4y ago
    > in math 1+1 = 2. That's an axiom That's not an axiom though. But (depending on your formal system of choice - for example Peano arithmetic) you can usually prove this using your axioms. What about "a+b = b+a" though? You can prove this in usual peano arithmetic, but there are (well-defined and studied by mathematicians) models, where you can prove that 1+1=2 but not that "a+b = b+a" (for arbitrary a and b). You could say that the truth of this statement is subjective. [1] A more famous example is an axiom of choice - you can decide that you use it or not, and you'll do a mathematic in a bit different universe depending on that. So I'd say there is more than "right" way of doing math. [1] https://en.wikipedia.org/wiki/Robinson_arithmetic https://en.wikipedia.org/wiki/Robinson_arithmetic
  - mannykannot 4y ago
    1 + 1 = 2 is a definition of 2, as the successor of 1. It is not a particularly fundamental one, as we can do arithmetic without it, just not in any base > 2. Definitions are axioms, and they are inescapable in philosophy. Rationality cannot get you any further than what follows from your axioms.
- Koshkin 4y ago
  How to add two whole numbers is a philosophical question?
  - msm_ 4y ago
    You have to define what a "whole number" is first, and that is a philosophical question. You may think it's obvious [1] what a "whole number" is and how it behaves under addition, but to make it formal you have to write it down. Pretty philosophical if you ask me. [1] everyone though that, until a few smart logicians showed up in XXth century and proved everyone else wrong.
RagnarD 4y ago
Every area of human cognition is affected by philosophy. Mathematics certainly merits its own philosophic treatment. A simple yet important example is treatment of integers vs. real numbers. In reality, integers map to the counting of discrete entities. Reals map to measurement of the attributes of entities. 'Entity' and 'Attribute' are already in the philosophic realm because they're fundamental concepts. These are important and valid distinctions. Far more can be said about these kind of philosophic classifications, and better understanding of them will lead to a better grasp of the connections between mathematics and the real world.
- bmitc 4y ago
  I loosely define philosophy as a field that is attempting to clarify and define what the questions actually are, that is in addition to providing frameworks and tools for which answers can sit within. Mathematics and the sciences operate in domains where the questions can be more readily known or written down and the basic tools exist already to attempt to address them. Any further development of new tools typically makes use of existing ones.
- thom 4y ago
  But at the lowest level, isn’t everything (time, space, energy etc) basically discrete, so the reals aren’t really real?
  - Koshkin 4y ago
    Just like integers, the real numbers are an abstraction, and so their reality as such is a philosophical question (which, despite being quite straightforward to answer, I have seen to confuse a lot of people).
  - yung_steezy 4y ago
    It is conceivable that the universe could be contiguous. Our current scientific thinking points to it being constituted from discrete pieces though.
  - DougMerritt 4y ago
keithalewis 4y ago
A philosopher's tools are paper and pencil. A mathematician's tools are paper, pencil, and wastebasket.
crazygringo 4y ago
If anyone's interested in the philosophy of mathematics, Wikipedia has a great overview: https://en.wikipedia.org/wiki/Philosophy_of_mathematics https://en.wikipedia.org/wiki/Philosophy_of_mathematics People have done a lot of thinking on this stuff over the centuries/millennia.
DubiousPusher 4y ago
> The move from ‘two numbers have the same same sum whichever way round you add them’ to e.g. ‘\forall x\forall y (x + y = y + x)’ changes the medium but not the message. And the fact that you can and should temporally ignore the meaning of non-logical predicates and functions while checking that a formally set-out proof obeys the logical rules (because the logical rules are formalized in syntactic terms!), doesn’t mean that non-logical predicates and functions don’t any longer have a meaning! Being able to use informal language to loosely describe something which also has a formal description is not Platonism. In fact, the only reason the statement "a sum is the same regardless of which way you order the arguments" functions as a language or math is because most people learn mathematics on the same set of real numbers and they take the domain for granted. In fact, I'd hazard a guess that most people don't even know there are other domains. This is not the secret operation of a world of "true" ideals about adding things. It's simply implicit information. In my journey through academia I would say I encountered many Platonist type thinkers in math and CS departments. There was definitely a popular notion that math contained some capital T, Truth. I have always been a bit suspicious of this notion. I encountered my first real critique of it in a work of pop philosophy of all places, 'Zen and the Art of Motorcycle Maintenance'. A theme of the book is criticism of the kind of "cult of reason" which operates in most universities. Particularly there is a section about mathematics. Pirsig's contention is that math is useful and it happens to be useful because we have invented it to be so. Math appears without contradiction because we invented it to not contain contradiction. We create these base assumptions and then reject anything that creates a contradiction. Pirsig points to the non-Eucludian geometry of Poincare as an example where we can drastically change the nature of the math itself by simply changing the base assumptions. Suddenly, what was contradictory, is no longer contradictory. For me this really solidified in my mind when learning Linear Algebra. Linear Algebra is all about laying out these 6 odd rules about vectors and scalars and then seeing where they take you and where they don't for that matter. Poincare's math turned out to have applications but it didn't have to. There are many dead ends in the world of high mathematics. And so Pirsig's ultimate observation can be boiled down to, Math is not true but it can be useful. This would later be echoed to me when I first heard the now popular saying amongst data scientists, "all models are false but some models are useful." This is how I've come to view mathematics. It's a giant pile of really well thought out and organized, often interconnecting models. Some of them match certain natural phenomena well. Others not at all. Some of them are obviously useful. Some of them are only discovered to be useful long after their invention (quaternions). Some of them may never prove useful. Ultimately I'd answer any contentions of math as model building and the author's tribulations about language here with this same observation. What happens when you want to take seemingly unrelated branches of math and connect them? What do you reach for? Set builder notation. x(weird E thing)(two legged R) blah blah blah for all of your constructs and then you start operating. You start by laying out all your assumptions in symbolic language. Then you let loose with the consequences of those definitions. Which sure sounds a lot like what modellers do in English when they're first proposing a model. The grand unifying layer of mathematics is a kind of symbolic modeling.
bmacho 4y ago
No, it does not, in literally any possible sense and meaning. You're welcome.
BLO716 4y ago
philosophy -> math -> computer science -> engineering. That's the traditional hierarchy from an educational perspective on one end to the other, as far as I visualize it. Everything has to start in the brain and be cognitively teased out to become the [math] part. The traditional scientific method then realizes it within the scope of the math foundation, that then in my world gets manifested into a software realization that can do mechanical things, and then it drives the physics and rubber-meets-road physical texture of it all. The realization of science always starts with a cognitive idea and general philosophy of one's ability. As Sheldon would say, "I have a general working knowledge of everything in the Universe."
kerkeslager 4y ago
Given the position of many philosophers that they're exempt from basic standards of evidence, I'm pretty done with those who call themselves philosophers. There's lots of very good philosophy being done right now, but the people doing it tend to call themselves scientists, psychologists, sociologists, anthropologists, etc. If you disagree, consider the field of ethics. Most ethicists taught in an ethics class immediately fall apart when exposed to the real world. The categorical imperative for example: if everyone become a computer programmer, we'd have no farmers and we'd all starve to death, therefore becoming a programmer is unethical! I have no problem teaching Kant's ideas if it's in the context of "this is what people historically believed" but that's not how it's taught in philosophy classes. Instead, they teach it as one idea of a few different ethical philosophies which all are valid and might be true. Because, you know, looking at the real world is the domain of science. This is philosophy, we take Kant Very Seriously. The predictable result of this is that self-described philosophers are the source of some truly abject nonsense. For example, consider the idea that free will could originate from quantum mechanics. Neuroscience has a pretty good idea of how decisions are made at the structural level, and the gaps are largely in the big picture of how the small pieces are coordinated, not how the small pieces (such as neurons) themselves, work. At the electrochemical level, the randomness or unpredictability of subatomic particles is probablistically eliminated for all practical purposes. As it turns out, the experimental conditions that allow us to demonstrate things like Heisenberg's uncertainty principle are hard to create, and don't occur in the human brain. This is obvious to quantum physicists, to the point that when I've asked quantum physicists about it, I've gotten the QP equivalent of "I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." But that hasn't in any way stopped philosophers from taking this idea Very Seriously. Mathematics doesn't need philosophy, at least not anything that has to be called philosophy. As the author notes, mathematics does have some implications for what the author calls "Big Picture questions", but inserting philosophers into the equation just confuses things. The author rightly notes that mathematicians are writing and talking about these questions, but what the author fails to note is that these questions are perfectly fine being part of the field of mathematics. Inserting the baseless opinions of philosophers with not expertise in math doesn't contribute anything--it actively detracts. I'll respond to an obvious criticism of what I'm saying: the author notes that many philosophers also have advanced degrees in math. But that just means that those philosophers also might happen to be adept mathematicians and might be able to answer Big Questions that happen to be mathematical. It doesn't mean that all philosophers, including those with no expertise in math whatsoever, suddenly have cart blanche to have opinions on mathematics.
User23 4y ago
The very effort of figuring things out, whether it's the behavior of natural systems or the properties of purely mathematical objects rests on the usually unspoken premise that things can in fact be figured out in some meaningful sense. That there are patterns to be found, that similar scenarios will play out similarly, and so on. Trying to prove that mathematically or scientifically isn't valid because it's begging the question. So instead we take the metaphysical position that yeah these things do have an underlying structure and that motivates us to try and figure out what that is.
chobytes 4y ago
Classic problem of trying to draw definitive lines between these things. I would guess this is more motivated by the social aspects of academics today than any real issue. In any event reading the work of philosophers has certainly made me a better thinker and mathematician. Without a broad base of thought one can easily get sucked into many intellectual traps and tar pits. In my case, I found Penelope Maddy and Wittgenstein to be helpful when I was figuring out what math "is".
m3kw9 4y ago
For things like understanding infinity maybe
_a_a_a_ 4y ago
Mathematics is a branch of philosophy. ISTM anyway. Always thought so and it just seems obvious.
- Koshkin 4y ago
  How to add two whole numbers is not a question of philosophy.
  - _a_a_a_ 4y ago
    Another pointless strawman comment. What a number is, what a negative number relates to, what addition is - all need making sense of. And numbers are more than addition. And maths is more than numbers (ponder logical implication). Consider the relationship between maths and the real world. etc etc. You might want to read up on the history of maths, it's likely odder than you'd imagine.
    - Koshkin 4y ago
      I was only objecting to math being called a branch of philosophy.

102 points by furcyd 4y ago

I don't think any philosophy is comprehensive enough to include all of mathematics and any partially complete choice is going to be subjective. You could even replace math with "x" and come to the same conclusion.

bugbuddy 4y ago

Betteridge's law of headlines dictates that the answer must be no .

pixl97 4y ago
Hence proving Betteridge's law is not universal.

monero-xmr 4y ago

I can't imagine a universe where 1 + 1 = 2 is incorrect. No matter what the constants are of any universe, in the abstract having 1 of something added to 1 of something is 2 of somethings. That is pure logic. If you can establish the axioms of zero, 1, adding, multiplication, etc., you can then continue into the rest of mathematics and also computation. As a side note I don't believe in infinity as anything other than a mathematical concept nor do I believe in parallel universes. I do believe there is a finite amount of universes (if more than one universe exists) and therefore a finite amount of matter. I think if we found intelligent life, they would have discovered precisely the same mathematical facts. To circle back to the philosophy of mathematics - pure math is pure logic. I think discovering God is more about physics and the Big Bang rather than via mathematics. However mathematical facts can be so pithy and perfect, that sometimes you wonder if God made them, and you are discovering mysteries of the universe as you find more mathematical discoveries. Discussing God on HN is not common, but given the linked post and its subject matter, it's hard to avoid that topic.

photochemsyn 4y ago
In symmetry, 1 + 1 = 1, where 1 (or c1) represents a 360 degree rotation around an axis. Also, 2 + 2 = 1, and 4 + 4 + 4 + 4 = 1 (180 degree rotation = 2, 90 degree = 4).
- monero-xmr 4y ago
  You are representing an abstract concept (circles) using base 360. That’s unrelated to the fundamental concept of adding 1 and 1 to get 2
  - ausbah 4y ago
    fundamental is just whatever axioms you choose, not your gut intuition of what "makes sense"
- Barrin92 4y ago
  mind you Platonism concerns the existence of mathematical entities, not their referents. If you take "1" to represent a natural number in one case and a rotation in another, that's not what matters. You could use hieroglyphics or emoji if you wanted. The question is, is the mathematical object and relation itself real? Formalists claim that mathematical expressions represent nothing at all, they're just syntax. The platonic question is, is there a mathematical symmetry where the identity operation does not hold? Is there Euclidian mathematics where a 360 degree rotation does not return you to your original state, where one thing is not equal to itself, and so on.
  - DiscourseFan 4y ago
    I mentioned this in another comment, but Platonism only is concerned with how humans are able to communicate with each other using shared concepts, if the "entities" exist as you say, most people who study philosophy would agree that insofar as they exist, they exist in people's heads (since, something existing in your mind is still real in so far as it has a material reality).

hackandthink 4y ago

Mathematics has a philosophical bent. Philosophy tries to make implicit understanding explicit and usually fails in interesting ways. (A part of) Mathematics tries to get the essence of numbers, calculation or the continuum and formulates corresponding axioms. So Mathematics tries to make the implicit understandig of for example natural numbers explicit. Mathematics doesn't fail usually because the scope is much smaller. I think some Mathematicians whose scope is very wide should be regarded as philosophers. Per Martin Löf: "Martin-Löf type theory is a formal logical system and philosophical foundation for constructive mathematics" (1) William Lawvere: his topos theory amalgates Logic and Geometry and leads to a kind of natural philosophy (2) (1) https://plato.stanford.edu/entries/type-theory-intuitionistic/ https://plato.stanford.edu/entries/type-theory-intuitionisti... (2) https://ncatlab.org/nlab/show/William+Lawvere#RelationToPhilosophy https://ncatlab.org/nlab/show/William+Lawvere#RelationToPhil...

alwayslikethis 4y ago
> Philosophy tries to make implicit understanding explicit and usually fails in interesting ways. > Mathematics doesn't fail usually because the scope is much smaller. Another way to look at this: we try to make implicit understanding explicit in a large number of fields. The fields in which it succeeds we call mathematics, and the other fields in which it fails we call philosophy.
- srcreigh 4y ago
  This hits to the core of CS, actually. Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable. Plug for implicit understanding though. Your very claim is not explicable; you can’t explicate the non existence of explicability. You can’t formally denote areas for which there’s no formal descriptions. This is the escape hatch out of logical realism (and yeah, that’s not an explicable claim, either; how fun!)
  - blackbear_ 4y ago
    > Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable. I am not sure that this is the case actually. Many results in mathematics only relate to the (non) existence of certain objects, without explicit recipes for finding (i.e., computing) them. And obviously philosophers started arguing whether an object that cannot be found actually exists or not :)

burna_aws_acct 4y ago

Maybe it's the directionality of the question that is confusing. How about, Does Philosophy needs Mathematics? To which I would answer, yes.

DiscourseFan 4y ago
I would disagree, simply because math deals with a fairly restricted sphere of inquiry, whereas philosophy might ask questions about things that mathematical logic, being restricted axiomatically, can't necessarily touch upon. But in truth, we should attempt to be interdisciplinary, I think when people work together on things and try to share their knowledge across field the most progress is made.
marginalia_nu 4y ago
This is as old as Plato though. His academy supposedly had a sign saying "let no man ignorant of geometry enter here". But it is very much bidirectional. Philosophy is necessary to understand the foundation on which mathematics sits. If you don't understand concepts like truth and existence, can you really meaningfully engage with work that deals with demonstrating truth and proving existence? The problem of universals is also very relevant as the article suggests. When prodded, you will probably be able to infer answers for what it means to exist, or what makes something true, but those are philosophical standpoints. They have arguments and counter-arguments. They imply an entire world-view. It's fine to make a philosophical standpoint, I think it's more questionable to do so without realizing their context, or even that you're even doing so as is often the case in the natural sciences. If you haven't studied philosophy, you tend to be very blind to the philosophical assumptions you and everyone around you is making. This is especially a problem as STEM education is absolutely steeped in philosophical assumptions and positions without offering the language or tools to recognize and discuss them, so they sort of seep into your mind through some sort of cultural osmosis. I think it's often perceived as though these standpoints (e.g. logical positivism) are the somehow the result of science, or inherent in the scientific process, but that's just not so. They're a means of interpreting the results, one of many.
- kerkeslager 4y ago

known 4y ago

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DiscourseFan 4y ago

I have a great respect for this author, but I do have to note that the debate of "constructivism" vs. "platonism" doesn't really exist in Philosophy per se, because it relies on a kind of basic misunderstanding of Plato in general. At least since Kant, the general consensus in Philosophy about Plato's "forms" or ideal objects is that they are conceptual objects, famously Kant argued that while reality in-itself is inscrutable, the means of judgement whereby we organize the world into concepts (beginning with the intuitions of time and space) are not, and by analysis of judgement we can begin to better understand the process of reasoning and therefore more rigorously understand the means by which empirical judgements are made (and thus further our understanding of empirical reality). Of course, many have attempted to answer the question of the psychological/biological explanation for how humans have shared concepts like this and shared, intelligible language. The most prominent thinkers of the 20th century, on both "sides" if you will, were Lacan on the "continental" end, and Chomsky on the "analytic" side. (Even though most Anglo-American--i.e. Analytic--philosophers would not accept Chomsky's logic as valid, and Lacan often referenced Anglo-American philosophy in his work...the categories are basically meaningless at this point.) But there are many other names, many people asking the above question, and I will not say who's thought attracts me the most, but I will say that most people I meet who study math are not familiar with them or the intellectual trends they are a part of, and they do not have a rigorous enough understanding of what philosophy is, really, to make claims about the usefulness of philosophy for their work--or even debate it, for that matter.

techno_tsar 4y ago
I believe that "platonism" in this article refers to mathematical platonism, i.e. the view that mathematical entities actually exist, as opposed to mathematical formalism, i.e. the view that mathematical entities are made up. https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat... https://iep.utm.edu/mathplat/ https://iep.utm.edu/mathplat/ This view is only called platonism because it parallels Plato's thoughts, but it is not an explicit reference to Plato's original philosophy. Modern platonism (after Kant) exists as an attitude about a kind of non-mental realism of abstract objects, and so it is with mathematical platonism. These days, there are very few Platonists (those who subscribe to the philosophy of Plato), but many platonists. As for the second point: A PhilPapers survey showed that most professional philosophers of mathematics are platonists, and most mathematicians who are familiar with the debate lean towards platonism. It is much more intuitive to understand how mathematical claims are true if you believe that numbers are real objects in the fist place, which in turn allows us to use the same accounts of truth we would normally use for non-abstract objects. At least prima facie, mathematical truths like 2+2=4 do not seem like they are different types of truth than empirical truths like "Biden is currently president". If mathematical objects like 2 and 4 weren't real, we would have to come up with a new account of truth that applies specifically to mathematics to explain what makes its statements true, which while possible, is a lot less elegant. Obviously, that doesn't refute platonism's alternatives, but in the words of a friend, it would really suck if he spent his career studying something that was entirely fictional, so he better be a platonist.

js8 4y ago

I have a good example why practical mathematics needs philosophy. In mathematical analysis and computer science, natural numbers and infinities have different character. In most of mathematics (like number theory or analysis), it's the countable infinite objects and their properties that are being studied, like natural numbers or infinite sequences. The finite sets themselves are considered somewhat trivial (natural) objects, and uncountable sets are mostly a universe where things happen. As a consequence, for mathematicians, all natural numbers are the same, no matter how large. In computer science, I would argue, the view is quite different. The interest of computer science is in medium-sized but finite sets (roughly between 2^6 and 2^(2^6)). The small sets of size less than 2^6 are trivial (you can brute force them without a computer), while with sets that are larger than 2^64 we almost don't need to compute with (they are like limits what can a DB store), and numbers above 2^(2^64) are pretty meaningless. So you can see, different natural numbers matter differently, some are important and some are less, they are not treated as identical objects. Since computer science is kinda part of mathematics, you can see the philosophical focus is different.

bibanez 4y ago
Still, what you prove for all naturals in Math still works for a finite number of them so the limitations in CS are just arbitrary, I don't think they matter much philosophically
- js8 4y ago
  They matter in what problems each field focuses on. And the idea that the natural numbers are all the same, but infinities have different character can be seen as arbitrary as well (cf. set theory).
  - ogennadi 4y ago
    We need INFINITY because no matter how much storage we have, we'll always have problems that need even more storage to solve?
srcreigh 4y ago

btilly 4y ago

My controversial take. Should mathematics prove to be in need of a philosophical insight, it will not come from a philosopher. The reason is simple. Most fields of study only find themselves in need of philosophical insights when they are internally driven to crisis. But when this happens, it is those who are part of said crisis that have the best insight as to the actual needs revealed, and it is from them that critical ideas come. Even if a philosopher happens to make a good point in advance, the philosopher will be ignored until the field of study has driven itself in crisis. After the crisis has passed, the philosopher's insight may be discovered to be relevant. But it is unlikely to have been noticed at the time. This is not just an abstract theory. It is borne out by history. For example calculus as envisioned by Newton and Leibniz had foundations in handwaving and wishful thinking. Bishop George Berkeley was absolutely right to complain about "ghosts of departed quantities". But it didn't become a crisis until the following decades when Joseph Fourier added together a bunch of sin and cosine terms (all well-behaved functions) and came up with a square wave (emphatically NOT a function by then current standards). While Fourier proved Berkeley right, Berkeley's complaint did nothing to help solve the crisis at hand. Instead we followed a decades long path through Cauchy defining infinitesmals in terms of sequences approaching zero, and then Weierstrass came up with limits when that followed. We dutifully put a passing mention to Berkeley in a lot of textbooks for being so prescient, but he was otherwise irrelevant. And so it proved again at the end of the 1800s. As mathematics again had a foundational crisis we came up with the three basic schools of mathematical philosophy that are still discussed. Platonism is a description of what mathematicians believed by default, though those who officially hold to it tend to be religious like Kurt Gödel. Formalism was essentially created by David Hilbert, a mathematician. Its opponent, Intutionism (which falls under Constructivism), was put forth by L. E. J. Brouwer - another mathematician. And so it has proven in other fields. When physics had a philosophical crisis with interpreting quantum mechanics, most of the important interpretations were produced by physicists. When evolution theory had foundational crises around how to incorporate genetics, most of the critical contributors to The Modern Synthesis were biologists. (Though one of the critical early contributors was R. A. Fisher, a statistician.) But philosophers were conspicuous by their absence. All that said, the people who come up with those key philosophical insights are often personally interested in philosophy. Having a mind that is capable of such philosophizing does seem correlated with curiosity about philosophy. BUT engagement with subject expertise during the period of crisis seems to be the more important criterion for important philosophical contributions.

richardanaya 4y ago

Mathematicians always have a philosophy. The only question is whether they know themselves well enough for it to be explicit.

n4r9 4y ago
I'm inclined to agree, and to extend the statement to physicists as well. Sometimes I even wonder if they're being disingenuous, for example Stephen Hawking on the one hand says "Philosophy is dead" and on the other hand actively engages in the debate over philosophical realism: https://medium.com/paul-austin-murphys-essays-on-philosophy/stephen-hawkings-model-dependent-realism-2b2973af41a6 https://medium.com/paul-austin-murphys-essays-on-philosophy/...
- GoblinSlayer 4y ago
  Maybe he thought philosophy is an attempt to figure out cosmology from pure thought.
Koshkin 4y ago
Philosophically speaking, everything is (or can be) a subject of philosophy, including philosophy itself.

deleted 4y ago

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practal 4y ago

I think philosophy is relevant to mathematics in the sense that one sometimes must be willing to step back from what one is doing, and ask deeper questions: Am I making assumptions here that are actually not true? This is a difficult thing, because how can you free yourself from assumptions you don't even know you made? Platonism can help here, because if you are trying to create a thing you know should exist, but you fail to do it, although you try again and again and again, then at some point you can either give up, because you think it is all arbitrary anyway, so why bother? Or you just keep digging, reevaluating your assumptions until something clicks.

guender 4y ago

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wslh 4y ago

But... mathematics has already a philosophy.... see Thomas Tymoczko [1] and traverse the links ;-) A great compilation is "New Directions in the Philosophy of Mathematics" [2] [1] https://en.wikipedia.org/wiki/Thomas_Tymoczko https://en.wikipedia.org/wiki/Thomas_Tymoczko [2] https://www.amazon.com/Directions-Philosophy-Mathematics-Thomas-Tymoczko/dp/0691034982/ https://www.amazon.com/Directions-Philosophy-Mathematics-Tho...

kerkeslager 4y ago
"Philosopher says math is philosophy" isn't proof that math is or needs philosophy.

callesgg 4y ago

Math is a form of and a part of philosophy. If you can’t see that, you are the one who needs philosophy.

Pinegulf 4y ago

Isn't mathematics just form philosophy where axioms are accepted?

nigamanth 4y ago
There are no axioms in philosophy, in math 1+1 = 2. That's an axiom. Within philosophy however, there's not one type of philosophy that is right, it's subjective so no.
- msm_ 4y ago
  > in math 1+1 = 2. That's an axiom That's not an axiom though. But (depending on your formal system of choice - for example Peano arithmetic) you can usually prove this using your axioms. What about "a+b = b+a" though? You can prove this in usual peano arithmetic, but there are (well-defined and studied by mathematicians) models, where you can prove that 1+1=2 but not that "a+b = b+a" (for arbitrary a and b). You could say that the truth of this statement is subjective. [1] A more famous example is an axiom of choice - you can decide that you use it or not, and you'll do a mathematic in a bit different universe depending on that. So I'd say there is more than "right" way of doing math. [1] https://en.wikipedia.org/wiki/Robinson_arithmetic https://en.wikipedia.org/wiki/Robinson_arithmetic
- mannykannot 4y ago
  1 + 1 = 2 is a definition of 2, as the successor of 1. It is not a particularly fundamental one, as we can do arithmetic without it, just not in any base > 2. Definitions are axioms, and they are inescapable in philosophy. Rationality cannot get you any further than what follows from your axioms.
Koshkin 4y ago
How to add two whole numbers is a philosophical question?
- msm_ 4y ago
  You have to define what a "whole number" is first, and that is a philosophical question. You may think it's obvious [1] what a "whole number" is and how it behaves under addition, but to make it formal you have to write it down. Pretty philosophical if you ask me. [1] everyone though that, until a few smart logicians showed up in XXth century and proved everyone else wrong.

RagnarD 4y ago

Every area of human cognition is affected by philosophy. Mathematics certainly merits its own philosophic treatment. A simple yet important example is treatment of integers vs. real numbers. In reality, integers map to the counting of discrete entities. Reals map to measurement of the attributes of entities. 'Entity' and 'Attribute' are already in the philosophic realm because they're fundamental concepts. These are important and valid distinctions. Far more can be said about these kind of philosophic classifications, and better understanding of them will lead to a better grasp of the connections between mathematics and the real world.

bmitc 4y ago
I loosely define philosophy as a field that is attempting to clarify and define what the questions actually are, that is in addition to providing frameworks and tools for which answers can sit within. Mathematics and the sciences operate in domains where the questions can be more readily known or written down and the basic tools exist already to attempt to address them. Any further development of new tools typically makes use of existing ones.
thom 4y ago
But at the lowest level, isn’t everything (time, space, energy etc) basically discrete, so the reals aren’t really real?
- Koshkin 4y ago
  Just like integers, the real numbers are an abstraction, and so their reality as such is a philosophical question (which, despite being quite straightforward to answer, I have seen to confuse a lot of people).
- yung_steezy 4y ago
  It is conceivable that the universe could be contiguous. Our current scientific thinking points to it being constituted from discrete pieces though.
- DougMerritt 4y ago

keithalewis 4y ago

A philosopher's tools are paper and pencil. A mathematician's tools are paper, pencil, and wastebasket.

crazygringo 4y ago

If anyone's interested in the philosophy of mathematics, Wikipedia has a great overview: https://en.wikipedia.org/wiki/Philosophy_of_mathematics https://en.wikipedia.org/wiki/Philosophy_of_mathematics People have done a lot of thinking on this stuff over the centuries/millennia.

DubiousPusher 4y ago

> The move from ‘two numbers have the same same sum whichever way round you add them’ to e.g. ‘\forall x\forall y (x + y = y + x)’ changes the medium but not the message. And the fact that you can and should temporally ignore the meaning of non-logical predicates and functions while checking that a formally set-out proof obeys the logical rules (because the logical rules are formalized in syntactic terms!), doesn’t mean that non-logical predicates and functions don’t any longer have a meaning! Being able to use informal language to loosely describe something which also has a formal description is not Platonism. In fact, the only reason the statement "a sum is the same regardless of which way you order the arguments" functions as a language or math is because most people learn mathematics on the same set of real numbers and they take the domain for granted. In fact, I'd hazard a guess that most people don't even know there are other domains. This is not the secret operation of a world of "true" ideals about adding things. It's simply implicit information. In my journey through academia I would say I encountered many Platonist type thinkers in math and CS departments. There was definitely a popular notion that math contained some capital T, Truth. I have always been a bit suspicious of this notion. I encountered my first real critique of it in a work of pop philosophy of all places, 'Zen and the Art of Motorcycle Maintenance'. A theme of the book is criticism of the kind of "cult of reason" which operates in most universities. Particularly there is a section about mathematics. Pirsig's contention is that math is useful and it happens to be useful because we have invented it to be so. Math appears without contradiction because we invented it to not contain contradiction. We create these base assumptions and then reject anything that creates a contradiction. Pirsig points to the non-Eucludian geometry of Poincare as an example where we can drastically change the nature of the math itself by simply changing the base assumptions. Suddenly, what was contradictory, is no longer contradictory. For me this really solidified in my mind when learning Linear Algebra. Linear Algebra is all about laying out these 6 odd rules about vectors and scalars and then seeing where they take you and where they don't for that matter. Poincare's math turned out to have applications but it didn't have to. There are many dead ends in the world of high mathematics. And so Pirsig's ultimate observation can be boiled down to, Math is not true but it can be useful. This would later be echoed to me when I first heard the now popular saying amongst data scientists, "all models are false but some models are useful." This is how I've come to view mathematics. It's a giant pile of really well thought out and organized, often interconnecting models. Some of them match certain natural phenomena well. Others not at all. Some of them are obviously useful. Some of them are only discovered to be useful long after their invention (quaternions). Some of them may never prove useful. Ultimately I'd answer any contentions of math as model building and the author's tribulations about language here with this same observation. What happens when you want to take seemingly unrelated branches of math and connect them? What do you reach for? Set builder notation. x(weird E thing)(two legged R) blah blah blah for all of your constructs and then you start operating. You start by laying out all your assumptions in symbolic language. Then you let loose with the consequences of those definitions. Which sure sounds a lot like what modellers do in English when they're first proposing a model. The grand unifying layer of mathematics is a kind of symbolic modeling.

bmacho 4y ago

No, it does not, in literally any possible sense and meaning. You're welcome.

BLO716 4y ago

philosophy -> math -> computer science -> engineering. That's the traditional hierarchy from an educational perspective on one end to the other, as far as I visualize it. Everything has to start in the brain and be cognitively teased out to become the [math] part. The traditional scientific method then realizes it within the scope of the math foundation, that then in my world gets manifested into a software realization that can do mechanical things, and then it drives the physics and rubber-meets-road physical texture of it all. The realization of science always starts with a cognitive idea and general philosophy of one's ability. As Sheldon would say, "I have a general working knowledge of everything in the Universe."

kerkeslager 4y ago

Given the position of many philosophers that they're exempt from basic standards of evidence, I'm pretty done with those who call themselves philosophers. There's lots of very good philosophy being done right now, but the people doing it tend to call themselves scientists, psychologists, sociologists, anthropologists, etc. If you disagree, consider the field of ethics. Most ethicists taught in an ethics class immediately fall apart when exposed to the real world. The categorical imperative for example: if everyone become a computer programmer, we'd have no farmers and we'd all starve to death, therefore becoming a programmer is unethical! I have no problem teaching Kant's ideas if it's in the context of "this is what people historically believed" but that's not how it's taught in philosophy classes. Instead, they teach it as one idea of a few different ethical philosophies which all are valid and might be true. Because, you know, looking at the real world is the domain of science. This is philosophy, we take Kant Very Seriously. The predictable result of this is that self-described philosophers are the source of some truly abject nonsense. For example, consider the idea that free will could originate from quantum mechanics. Neuroscience has a pretty good idea of how decisions are made at the structural level, and the gaps are largely in the big picture of how the small pieces are coordinated, not how the small pieces (such as neurons) themselves, work. At the electrochemical level, the randomness or unpredictability of subatomic particles is probablistically eliminated for all practical purposes. As it turns out, the experimental conditions that allow us to demonstrate things like Heisenberg's uncertainty principle are hard to create, and don't occur in the human brain. This is obvious to quantum physicists, to the point that when I've asked quantum physicists about it, I've gotten the QP equivalent of "I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." But that hasn't in any way stopped philosophers from taking this idea Very Seriously. Mathematics doesn't need philosophy, at least not anything that has to be called philosophy. As the author notes, mathematics does have some implications for what the author calls "Big Picture questions", but inserting philosophers into the equation just confuses things. The author rightly notes that mathematicians are writing and talking about these questions, but what the author fails to note is that these questions are perfectly fine being part of the field of mathematics. Inserting the baseless opinions of philosophers with not expertise in math doesn't contribute anything--it actively detracts. I'll respond to an obvious criticism of what I'm saying: the author notes that many philosophers also have advanced degrees in math. But that just means that those philosophers also might happen to be adept mathematicians and might be able to answer Big Questions that happen to be mathematical. It doesn't mean that all philosophers, including those with no expertise in math whatsoever, suddenly have cart blanche to have opinions on mathematics.

User23 4y ago

The very effort of figuring things out, whether it's the behavior of natural systems or the properties of purely mathematical objects rests on the usually unspoken premise that things can in fact be figured out in some meaningful sense. That there are patterns to be found, that similar scenarios will play out similarly, and so on. Trying to prove that mathematically or scientifically isn't valid because it's begging the question. So instead we take the metaphysical position that yeah these things do have an underlying structure and that motivates us to try and figure out what that is.

chobytes 4y ago

Classic problem of trying to draw definitive lines between these things. I would guess this is more motivated by the social aspects of academics today than any real issue. In any event reading the work of philosophers has certainly made me a better thinker and mathematician. Without a broad base of thought one can easily get sucked into many intellectual traps and tar pits. In my case, I found Penelope Maddy and Wittgenstein to be helpful when I was figuring out what math "is".

m3kw9 4y ago

For things like understanding infinity maybe

_a_a_a_ 4y ago

Mathematics is a branch of philosophy. ISTM anyway. Always thought so and it just seems obvious.

Koshkin 4y ago
How to add two whole numbers is not a question of philosophy.
- _a_a_a_ 4y ago
  Another pointless strawman comment. What a number is, what a negative number relates to, what addition is - all need making sense of. And numbers are more than addition. And maths is more than numbers (ponder logical implication). Consider the relationship between maths and the real world. etc etc. You might want to read up on the history of maths, it's likely odder than you'd imagine.
  - Koshkin 4y ago
    I was only objecting to math being called a branch of philosophy.