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.
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.
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.
>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.
Remember, fiction is real: it takes time to be written, its distributed to its audiences, it has a definite cultural impact and often changes how people view the world. Entirely fictional plays have spawned political revolutions--there are many forces at play in a fictional work that are certainly objective and real. There isn't anything that isn't "real", the question is not "is math real" (it definitely is), the question is "where does math exist?", and on account of its location (inside or outside the head), how we should be approaching it and where it should be applied. Nobody would say math isn't useful, but we might say it could be more useful if its basis is more rigorously examined.
As a (former) student of psychology, I personally subscribe to the view that both platonism and constructivism are true (edit: in that they both accurately depict different-but-interrelated aspects of mathematics).
It’s a false dilemma much like “is light a wave or a particle” or “free will or determinism”. (Yup, I am that “you can have it all” pollyanna type of person.)
I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.
They're not an approximation, a metaphor, or a point of view. They're the real deal - the base mechanisms. And they can be discovered through the scientific process, with its combination of physical and speculative analysis.
Clearly this is nonsense. Math isn't truly self-consistent, physical research is limited to a range of lab-friendly experiments garnished with some astrophysical guesswork, and all of it gets filtered through consciousness, which we have no clue about.
What we have is a "looking for the key under the light" situation where can only explore the things we can see. We don't know what's in the dark, and it's actually very likely that our consciousness is extremely limited and unable to perceive essential detail.
But because (tautologically) we can't see it we just assume it's not there, and our tiny and contingent view is gloriously universal.
I find the cat metaphor very revealing. Cats share a space with us but they literally do not see the same objects we do. They perceive weight, texture, and dimensions, they're far more sensitive to smell, and they have some innate models for dynamics and mechanics.
But they have no concept of the meaning of a book, a laptop, a wifi card, a Netflix subscription, or a mathematical description of General Relativity. Unless we breed them specially for intelligence for a good few tens of millennia they never will, because cat consciousness is too small to contain those concepts.
It's ridiculously, almost comically naive to believe - purely on faith - that human consciousness isn't severely limited in some analogous ways.
We're quite good at the human equivalent of hunting for food - which includes manipulating physical materials and crude energy sources, with some meta-awareness of abstraction.
What are the odds that's all there is to understand about the universe?
> I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.
Consider Euclidean geometry, which is fine, mathematically, but, it turns out, not how the universe works. If mathematical platonists were concerned about whether mathematics' forms are how the universe really works, surely they would insist on the verification of their axioms before proceeding? And then, would mathematical platonism not be just the uncontroversial parts of the physical sciences?
The author reprises the platonism / formalism issue in this article's final section, beginning with the paragraph "But there is still more to be said. Perhaps, after all some of those Big Picture questions do remain lurking in the mathematical background." He refers to Platonists as realists, but, I think, in the sense that the forms are real regardless of whether they are how the universe really works.
I would consider the reverse (that mathematics is constructed) as just as nonsensical. One could argue that if mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented. But this seems backwards. The Pythagorean theorem makes more sense as something we have discovered, or the inner corners of a triangle add up to 180 degrees (half a circle); alien civilizations likely have arrived at the same conclusion. The simple answer is that math is simply a feature of the universe.
Even if mathematics isn’t truly self-consistent (it is not), that does not commit one to formalism or constructivism. The belief that abstract entities, if they are real, must be self-consistent, requires us to believe that self-consistency is a precondition for the realism of abstract entities to begin with. But there is no obvious reason to believe this.
As for the limits of our human consciousness: arguably there is a “floor” where we can have strong beliefs in the hypotheses we form about the universe (including those of mathematics). In fact, the essence of Platonism (and where it derives it’s name) is the very view that abstraction is realer than concrete or empirical particulars because it is more unchanging and absolute. It seems inconceivable to find a single world where 2+2 != 4, but we can conceive of worlds where say, Biden is not currently president, or where gravity had a different strength. In other words, the laws of logic (and perhaps many parts of mathematics) seem very fixed, but our other laws less so. Plato thought this told us something about the ultimate hierarchy of metaphysics; modern mathematical platonists like Godel think that we have a mathematical intuition that allows us to perceive mathematical objects; mathematicists like Max Tegmark thought that nothing other than mathematical objects exist at all.
It is this intuition towards the abstract as real, realer, or realist that motivates platonism. To committed platonists, the burden of proof is actually on the non-believers, partly as a preservation of logic and mathematics. If we dismiss that (a common logic), we might be incapable of having a real discussion in the first place. Whether or not that intuition is enough, or free of problems (it is not) is very debatable. However, platonism is not trivially or obviously false.
> If mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented.
Err, no.
The mathematics we as human have constructed is a superset of the physical universe - the mathematical world embraces much more than the mere confines of the physical world we can kick and observe across.
Indeed we have constructed various mathematic worlds that are at odds with each other - some my have application in this physical universe which then precludes others from also corresponding to the same.
So to be a Pollyanna is to have a certain (overly?) (irrationally?) optimistic towards ones situation in life and perhaps even the nature of human suffering in general. To be contrasted with the Buddhist thought which asserts that basically life is suffering: https://en.wikipedia.org/wiki/Four_Noble_Truths
===
Anyway, back to your belief that (mathematical) Platonism and (mathematical Constructivism can be reconciled.
So a hard-core mathematical Platonist believes – if I am not mischaracterising their position – that things like numbers are actually existing entities that we discover and that even if we humans had never existed that numbers would have or that if we humans cease existing that numbers will continue to exist. So mathematical progress is a progress of discovery, not creation. A hardcore mathematical Constructivist believes that if it were not for us numbers would not exist, the very idea of number would be unthinkable, we think numbers into being. A radical Constructivist (like me) does not believe in actually existing infinities or infinitesimals and so does not believe in actually existing unbounded real numbers like irrationals and transcendentals except in a symbolic or algorithmic sense.
Can these two positions be reconciled? Can we optimistically reconcile them. I think not. But I do think that the slighter weaker proposition that certain formal entities like numbers are necessary, I'm going to say, "truths" in that the nature of the universe necessitates certain types of mathematical entities such that once there are a sufficient class of thinking things to think them they'll pop into being, so to speak.
I am open to correction on any point. I know that my personal position is more-or-less anathema to most mathematicians I've had the pleasure of sharing these ideas with (as in I've gotten into heated drunken debates/arguments about this stuff).
Wow, I guess my mental model (so to speak) is even more “radical” than yours. I don’t think mathematics is really part of the (empirical) universe, but that they are their own kind of abstract entity. They may happen to correspond to certain patterns in how things exist and interact in the “real world”, or to sentient beings’ reasoning and modeling faculties, but they are not tied to the real world either way.
For comparison: To me, for a number to exist in a “symbolic or algorithm sense” is to for it to exist, period - but in the sense of “creating”/“discovering” a new number system to contain them. The set of rational numbers isn’t really “special” to me. (Non-negative natural numbers are “special” for their association with cardinality, but I will refrain from going down that rabbit hole this time.) (I assume you meant “unbounded” in terms of expansion into elementary algebra; do correct me if I misunderstood you.)
(i.e. Existence=NaN because it’s a loaded word, Abstractness=Yes, Independence=I have some but limited sympathy for the neo-Fregean view on this, “creating” and “discovering” are the same thing to me)
(Which would make me a platonist to some people, an intuitionist to others, I guess)
Plato is worth reading in the original because of the way it makes one think. But we might call it Pythagorean if we want to claim that the world is made of Math.
Roger Penrose has a very nice “3 worlds theory” about mathematics, the physical world and the world of mental experience. He describes theories for how these worlds encompass each other and overlap. In “The Road to Reality.”
In addition to this, Plato is worth reading because he is still relevant and his work outlined the agenda of huge swathes of western philosophy, which undeniably influences our scientific paradigm and political worldviews.
I have not yet read The Road to Reality. At 1094 pages, I hope it is not extremely complicated.
Plato gets summarized — and that mostly doesn’t work. Need to read it. The Parmenides is supposed to be the hardest to understand—pah, I’d start with that. Cut to Zeno arguing conclusively that all is one. He then conclusively argues that everything can’t be one. Then…
Don’t be put off by the size of Penrose’s road to reality! The introduction and first chapter already made the book worth it. I’ve not gotten much further myself (but that’s how I read)
Plato can't be summarized, there are no conclusions to any of the arguments, just endless arguments. That's why he's so great, he always leaves the question open, there is always room for greater understanding.
2+2=4 is (roughly) the same kind of truth as “two groups each consisting of two elves have a total of four elves”, or “if you travel a distance of 2cm twice, you’ve traveled by 4cm”, except it isn’t tied to the real or fictional existence of centimeters or elves.
And loosely the same kind of truth as “imagine a world where elves live in Lorien… in this world, elves live in Lorien.”)
And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.
That's a tautology, you can similarly say "imagine a world where mathematics isn't tied to the real or fictional existence, in this world mathematics isn't tied to the real or fictional existence".
>And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things.
I didn't do such things.
> Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.
It's derived from initial assumptions, which is how all math works.
1. I would object to the “similarly”, because they are not similar types of statements. And yes, the tautology aspect is the whole point of the axiomatic method (which has limitations that cannot be directly blamed on that premise).
2. You didn’t do the first two. But the symbols now mean different things than their conventional interpretations in number theory.
3.
> It's derived from initial assumptions, which is how all math works
It’s exactly how _logic_ works, and is how all math works, but that would only qualify it as (il)logic, and not inherently math. Necessary but not sufficient condition.
> or all the symbols mean completely different things
It's actually not entirely unproductive to consider this line of thought, whereby in this formulation equality actually means something like "arrivable via some number of zero divisions". I'm sure you could find all sorts of curiosities with this mathematical "toy".
Yup. You run into that all the time in abstract algebra. Although people usually don’t like to touch the equality sign; the usual practice (based on my limited exposure) is to invent equivalent operator notations.
Exactly - it’s still there (and still has its own semantics), but it’s also inert.
And, to address your point directly, of course mathematics detached from context can be used in practice. One can certainly create-slash-discover an abstract algebra, derives theorems about it detached from outside context, and then later on discover a context in which the abstract structure is applicable, and apply the pre-derived theorems.
I actually agree with you - “the symbols mean something different now” isn’t a bug, it’s a feature. But I was trying to point out (what I saw as) a big ambiguity in parent’s comment.
I've talked to a fair number of mathematicians down through the years and the vast majority I've talked to are Platonists. Some have even thought about this stuff and know what Platonism means and assert that they are Platonists.
For me, once you read up on L.E.J. Brouwer's intuitionism and think through its consequences properly you can't be a Platonist, but that's just me – I undoubtedly have a prior more fundamental belief that makes me think this way. Like, I think it'd be difficult to be a (mathematical) Platonist and an atheist at one and the same time, but many mathematicians are.
Yeah, this seems to be among the hardest questions that mankind has to deal with, so it's probably not surprising that these definitions are often confused :
Don't these arguments often essentially become about the definition of "real" or "exists"? It seems to me that I can make mathematical Platonism true or false by using a weaker or stronger definition of "exists"
Being stirred by your last sentence, I have a lot of trouble reading some philosophy. I have read your comment several times, and I am still not sure what it is saying.
> 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.
What is the misunderstanding, exactly? Did you end up addressing this? The next sentence is a long run-on, and I'm getting lost in what you're saying Plato actually said versus what you think people think he said. And I think I agree with another commenter that you are confusing Platonic and platonic.
And what does philosophy say about constructivism? If it doesn't address it versus some notion of idealized objects existing, then why does that matter to mathematics? It is a legitimate line of thinking and isn't an exclusively philosophical subject.
> 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.)
There are just names and labels here. What are you saying here?
It's very weird that you would reach for Lacan and Chomsky. Neither of them are really seen as credible in the field of philosophy, nor have they had really large impacts in philosophy. Cultural criticism and linguistics? Sure, I guess. But not in philosophy. Even in the 20th century "continental" tradition of cultural criticism, Lacan is not very high in the eyes of many. Heidegger (controversially, of course) or Maurice Merleau-Ponty or Paul Ricoeur have been much more influential. For whatever reason, Lacanian analysis did burn bright in the 70s and 80s, but man has it fallen hard.
Arguably all modern philosophy of language in the 20th century was influenced by Ludwig Wittgenstein. Wittgenstein is, for many philosophers, the most influential philosopher of the 20the century. Wittgenstein's "method" and work in Philosophical Investigations also offers the most far reaching and consequential criticisms of platonism, cartesian mind-body dualism, and philosophical skepticism, in the philosophical canon.
I do agree with your closing comments though. Most people aren't very well versed in philosophy and operate on a sort of pop understanding of it. They simply lack the framework to make credible claims on the usefulness of it in general. It's especially hilarious when they try to say one of the longest and most fruitful intellectual disciplines in human civilization is "not very useful." Sure, bud.
> It's especially hilarious when they try to say one of the longest and most fruitful intellectual disciplines in human civilization is "not very useful." Sure, bud.
One of the longest? Sure. One of the most fruitful? I'm not so sure. What are the concrete, usable conclusions of philosophy, and what are the fruit of those conclusions? How do they compare to the fruit of the usable conclusions of Newtonian mechanics or Maxwell's Electrodynamics?
Unfortunately much of philosophy is very easily ignored. The philosopher will get very defensive about this, as perhaps they should, since their work was the "initial" work leading to the development of a rigorous field (physics, mathematics, psychology, etc). However, much of philosophy is intentionally vague because the philosopher is grasping at straws, is speculating.
Philosophy is a discipline whose success is desperately trying to reclassify itself from its own subject. In that regard, when you say that mathematicians do not care and cannot argue philosophy, you are merely being defensive. There is nothing in the philosophy of math that a Mathematician could not grapple with at your level of expertise or better in a simple conversation with no preparation. The issue is that what you have to say is not rigorous, it is not grounded in anything, it is vague. This is your problem, not the mathematicians.
> Kantianism is not the predominant view among working metaphysicians today
Which gladly nobody claimed to be the case. "At least since Kant" gives us a time frame not a causal link — and even if it did pretend there was a causal link, any philosopher wouldn't have to be a Kantian in order to agree to a thing Kant said about Plato's concept. Or phrased differently: A Non-Kantian can agree with Kant on a thing Kant said about Plato without suddenly becoming a Kantian.
What the poster said here is to my knowledge correct (I studied philosophy).
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.
> But at the lowest level, isn’t everything (time, space, energy etc) basically discrete, so the reals aren’t really real?
That literally doesn't matter one way or the other in the slightest, for two huge reasons:
1. No mathematics is exactly the same as reality itself, and cannot be, even in principle. Mathematics can be used to useful model certain limited aspects of reality, but of course models aren't the same as reality.
2. The mathematics of the reals and of continuity in general are essential for a vast number of kinds of practical computation that sometimes (not always!) turn out to be vastly harder to do without the assumption of continuity. We often want to bring to bear the whole apparatus of established mathematics to solve problems in order to get answers, and no one trying to get answers cares whether the mathematics used is "real", only whether the final answers are useful (accurate etc).
Actually there's a third reason that is even more important to some people: this is usually merely about word definitions. Words like "real" and "exist" have sharply defined unambiguous technical definitions in mathematics that are only indirectly related to the loosely defined vague non-technical definitions of the same words that we use in natural languages like English.
It is ultimately irrelevant whether e.g. the real numbers "exist" in the physical universe, because whether they do or not, they provably exist in (most) formal mathematics -- the word "exist" does not have the same meaning in the two realms.
It's still interesting to ask whether the physical universe has reals or continuity or infinities or infinitesimals etc., it just shouldn't be confused with the issue of their technical existence (or "reality") in mathematics.
This thinking has some superficial appeal, but I don’t think quantization in various domains frees you of the need for the continuum when those domains interact.
I don’t think there’s any reason to believe that, say, the fine structure constant, or the ratio of the mass of a neutrino to the mass of an electron, are rationals, or even algebraic.
Maybe there is though? That seems like very much a question for philosophy, since scientific measurement will never be able to tell us for certain.
In "reality", there is no such thing as an electron that has some mass. It's leaky abstractions and approximations all the way down. If we had a "complete" theory of physics, it would still be the same situation for all practical purposes, since you can never know "for sure" that is it "correct".
Hence, rationals are plenty sufficient for all of physics, assuming you bother to reformulate all the theories that are historically based on real numbers. However, there is no point in doing this, so only a few niche people try (who, of course, do it for philosophical/aesthetic reasons).
The tools of mathematics are actually just better suited to working on the continuum than on discrete numbers. So maybe that does suggest that when physicists use the tools of mathematics they will always wind up with real approximations to discrete reality.
No matter that physical reality insists that the number of people in a population or U-238 atoms in a lump of metal has to be an integer, mathematics will tell you that that integer nonetheless has a distinct, nonalgebraic natural logarithm, and that that number is useful in predicting how many people will be in that population or U-238 atoms will remain in that lump of metal at some later time - even if the raw result of any such calculation will be a nonsensical noninteger.
If the universe is infinite in size (given that measurement is close to the topology being flat), or time is infinite in the future, then no. Or there is an infinite multiverse (take your pick of which one).
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).
> Every area of human cognition is affected by philosophy. Mathematics certainly merits its own philosophic treatment.
This is the problem I have with philosophy. It's a huge category error. It's too broad. Why is there a philosophy of math and then a philosophy of ethics or art?
Art and ethics are clearly human experiences and really arbitrary things that are made up by people. A bug or a bird or a hyper intelligent space alien won't be familiar with these concepts because they are, in general, unique to humans.
Math on the other hand has the clear distinction of being more universal. The underpinnings of math and logic end up being hierarchically above religion and ethics because you can describe the entire universe using the principles of logic.
You have an atom, then from the rules of what describes an atom, you can build a neuron, then from the neuron the human brain, then from the human brain, religion and ethics. But the problem is philosophy places these things side by side as if they're in the same category. The emotions evoked by Monet's paintings have no place next to say a paper on quantum physics.
I feel philosophy just encompasses any topic we can talk "deeply" about. Because literally anything that can be analyzed with great depth is a philosophy... philosophy ends up basically becoming the study of anything on the face of the earth... which is a pointless category.
When you say "merits" philosophical treatment. It basically means merits deep thought and analysis. Why package it up as if it's actually a field with known technical methods of analysis instead of what's actually going on that is random deep musings with no quantitative rigor.
None of the conversations you mention can happen outside the context of an epistemological framework, yet epistemology is far from a settled field. Thus conversations in the aforementioned areas are often reduced to debates on the merits of the respective epistemological premises of different positions.
Epistemology is just another category error in a category error. What is the study of knowledge? It's the study of everything on the face of the earth, just like philosophy.
There are models of the teams that are countable when looking at the model externally within ZFC. However, that model thinks it is uncountable. When you say “real numbers” what exactly do you refer to? I know you mean the standard model but other models think they are real numbers too.
How do you know a better understanding of these kinds of philosophical classifications will lead to a better grasp of the connections between mathematics and the “real” world? What is the definition of real world that excluded mathematics?
He discusses two relatively simple proofs, the first being Euclid's proof of the infinity of the primes, the second being Pythagoras's proof of the irrationality of the square root of two:
> "Euclid’s theorem tells us that we have a good supply of material for the construction of a coherent arithmetic of the integers. Pythagoras’s theorem and its extensions tell us that, when we have constructed this arithmetic, it will not prove sufficient for our needs, since there will be many magnitudes which obtrude themselves upon our attention and which it will be unable to measure: the diagonal of the square is merely the most obvious example. The profound importance of this discovery was recognized at once by the Greek mathematicians. They had begun by assuming (in accordance, I suppose, with the ‘natural’ dictates of ‘common sense’) that all magnitudes of the same kind are commensurable, that any two lengths, for example, are multiples of some common unit, and they had constructed a theory of proportion based on this assumption."
> "Pythagoras’s discovery exposed the unsoundness of this foundation, and led to the construction of the much more profound theory of Eudoxus which is set out in the fifth book of the Elements, and which is regarded by many modern mathematicians as the finest achievement of Greek mathematics. The theory is astonishingly modern in spirit, and may be regarded as the beginning of the modern theory of irrational number, which has revolutionized mathematical analysis and had much influence on recent philosophy."
Another good discussion is Poincare's "Science and Hypothesis" (1902) in which he asserts that one defining feature of all mathematics is the self-consistency of arguments, which is the freedom from contradiction (a view all philosophy would hopefully adopt, vs. say, the expediency of political behavior and nation-state propaganda).
> "To sum up, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. The only limit to its power is the necessity of avoiding all contradiction; but the mind only makes use of it when experiment gives a reason for it."
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.
That's a pretty flimsy argument. Yes, you can map mathematical ideas onto concepts within different fields of study, but that's kind of the point. Mathematics aims to abstract away any such notions of where you might encounter the natural numbers or the reals, "in reality" or elsewhere. All you've persuaded me of here is that philosophy has a use for math.
> 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.
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!)
It has been striking to me how post-modern philosophy seems to have recentered itself around the questions of language... while «naming things» is considered one of the ~~two~~ three «hardest problems in CS» !
I’m not really bickering about names here, just trying to succinctly describe ideas.
The irony of the GGP comment is that it claims mathematical ideas are successful ones and philosophical ones aren’t, but that’s not a mathematical claim. So either it’s right and it’s wrong, or it’s wrong.
Not directly related, we're now surrounded by computers, which are even dumber (some forms of machine learning and analog computers aside ?) in that they have zero flexibility in treating symbols as something else...
> 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 :)
My favourite example this is Mark Braverman's demonstration that all Quadratic Julia sets are computable (as in computable real numbers: meaning that you can produce arbitrary fine approximations).
However, as Mark notes, this proof proceeds by deriving 5 alternative programs to do the computation. For any given Julia set parameter 'c' (given as a computable real number) one of the 5 programs will compute the Julia set. Which one? Who knows. It depends discontinuously on the properties of 'c'.
Proving non existence uses some type of law of excluded middle. It’s a very concrete and computable style of argument, although its subjects may not be.
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/...
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.
This view of CS seems to exclude two of the most important topics - undecidability and NP hardness.
the main techniques in both of these areas require infinite sets. For undecidability, it’s the infinite set of proposed halting problem TMs, and for NP hardness, it’s the infinite set of solutions to a given problem.
Runtime analysis also requires numbers larger than 2^64 — any algorithm which is not O(n) is practically faced with these numbers. Sorting, matrix multiplication, etc.
Yes you can have mathematical techniques in CS, I am talking about focus or emphasis of the discipline. Obviously, there are gonna be subjects on the fringes (set theory is another example).
Similar to what you say, mathematical analysis uses uncountable sets (especially the ones above reals). But they're just a theater, not the actors. The actors are the countable sets - sequences, continuous functions..
And I think similar situation is in CS, where infinity, while useful, is just a theater to uncover properties of medium-sized finite objects.
For many people, philosophy is kind of "obvious babbling". But as you can see, that's what we engage here. It's interesting to think about this in the context of Skolem's paradox - I think the two views (the mathematical and the CS emphasis on nature of numbers) are actually equally valid.
I disagree my friend. CS is math; they aren’t separate. The topics are at the core, not on the fringe. And if we need infinity to understand the small objects, then infinity is more important than the small objects themselves.
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
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).
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.
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.
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.
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.
If they are unfamiliar with natural numbers, they may believe natural number is an inconsistent concept and won't believe that 1+1=2. Similar to how people believed round earth is inconsistent.
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).
That's not true. In the Philosophy of Mathematics when people refer to Platonism they make two claims. That mathematical objects are 1. abstract and 2. independent. (also this is the case in traditional Platonism).
An object is abstract if it is not spatiotemporal or causal. Asking what the number 4 weighs or does makes no sense. It is not mental, i.e. mind independent, in that it exists outside of any agent's thoughts. A Platonist would argue that even an alien civilization is going to discover, not invent, logic and natural arithmetic. Unlike say, Inglourious Basterds which is a mental product that did not exist before Tarantino thought it up.
It is debatable if that is the case in traditional Platonism, the evidence for artificial forms comes from a single epistle (the 7th) that is arguably fabricated. I've read some of this epistle in the original, and stylistically it is very far removed from Plato's regular style, and the language resembles (to me) Koine Greek and not Attic Greek.
In any case, just because someone calls themselves a platonist doesn't mean they've carefully studied Plato. In the same fashion as Kant, asserting time and space precede the apprehension of objects that are necessarily within time and space does not imply that time and space are necessarily part of the world itself, but only that one's means of judgement begins with the notions of time and space. In the same way, whether or not you agree with Kant, it is entirely plausible to assume instead that there is a shared faculty in the minds of humans that allows us to conceive of things like 4, and addition and subtraction and mathematical functions in general, and this faculty would neither be abstract nor independent but biological and empirically observable. If this latter case is true, then it would prove to be far more useful for scientific investigation than simply assuming that the reason why we can communicate about the number 4 is because it exists abstractly and independently, because that doesn't tell us anything more about why we have shared concepts of numbers in the first place.
It seems misleading to call this "symmetry" because what you wrote is not a group.
The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).
But they might start with different axioms (e.g. non-euclidean), exploring in different directions, and perhaps discover truths new to us that affect our own mathematics.
By definition, this is hard to imagine; I mean, try to imagine a mathematics that isn't even based on numbers...!
Numbers are just this strange paint that humans apply to some things. Real computers are not based on numbers, they are based on physics of the natural world, which we only model with numbers. There is no ontological confirmation that reality is numerical; we just have a small handful of experiences which indicate that it can be understood as such.
This hits to the core question of whether math is real or fantasy. If we built a computer to calculate 1+1=2 and prints true for some millennium until one day when it suddenly prints false; what type of problem is that? Is math now wrong, or is it safe?
what if one day 1+1 becomes meaningless to us? The computer prints 2 but suddenly it’s as static to us? This seems like a downgrade, but what if this is accompanied by direct knowledge or an upgraded type of model to understand reality? It’s not seen as 2, but the underlying natural mechanisms are perceived directly? Math is broken, it doesn’t even seem to exist, yet the world continues on and we perceive it in some new strange predictive certainty?
The claim that the universe is fundamentally analog is on a somewhat shaky ground in postmodern (quantum) physics, especially once you bring (discretized!) information into it, which has to happen quite "soon" if you want to deal with statistically emergent phenomena like temperature.
They quantize because the underlying solution is a circular standing wave, which is stable if it consists of a whole number of periods. The wave itself is of course analog, quantization emerges through many hoops and provides stable states with certain potential well like any stable state, and can squiggle a little under perturbation or squiggle a lot under, say, Zeeman effect.
Can we really speak of the wave(let) "being" analog when it is impossible (according to our current understanding of physics) to see that experimentally ?
> I can't imagine a universe where 1 + 1 = 2 is incorrect
I think it really depends on lots of factor, like how their biology works, how their civilization works, how their perception works, what are their goals etc. There could be a civilization where they don't use "logic" (as we know it) as a tool to develop their civilization, maybe they have a form of logic where true can sometimes be false based on mood or state of the world (but of course we won't call that "logic" when we see it)
> I can't imagine a universe where 1 + 1 = 2 is incorrect
I can. For instance, I was playing Civilization one day, I thought my empire should have 32768 gold but apparently I was 32768 in debt! So such universe can certainly exist, especially if it's a simulation.
for me, yes, 1+1 = 2 is platonic. I start going all formalist when thinking about the continuum hypothesis [0]. The more I try to understand what's going on with the number and size of infinities, the more it seems that they're no longer something platonic, just something that drops out of a rule engine operating on a weird formal system.
Disclaimer: not a mathematician so it might be obvious to others that these things exist in platonic space.
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.
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".
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.
> This is as old as Plato though. His academy supposedly had a sign saying "let no man ignorant of geometry enter here".
Sure, but by that standard most modern philosophers wouldn't be allowed in. In Plato's time philosophy was the study of everything, whereas today philosophy is what's left over now that all the useful studies have split off into their own fields.
The article points out that some philosophers also have advanced degrees in math, but it's a pretty big leap from "Some philosophers have things to say about math because they're mathematicians" to "Philosophy has things to say about math".
> 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.
This is true, but why should I care what philosophers have to say about truth and existence? Math and science have plenty to say about truth and existence without philosophy's help.
You might call that philosophy, but that's just a semantic argument--it can just as reasonably be called math or science. Calling it philosophy is just a way for philosophers to try to get people to listen to their baseless opinions about fields they haven't gained expertise in. If I want to know what truth or existence mean in math, I'll talk to a mathematician, not a philosopher.
> 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.
Is it? What are some other ways of interpreting results?
Keeping in mind that "making shit up" isn't a way of interpreting results.
> This is true, but why should I care what philosophers have to say about truth and existence? Math and science have plenty to say about truth and existence without philosophy's help.
> You might call that philosophy, but that's just a semantic argument--it can just as reasonably be called math or science. Calling it philosophy is just a way for philosophers to try to get people to listen to their baseless opinions about fields they haven't gained expertise in. If I want to know what truth or existence mean in math, I'll talk to a mathematician, not a philosopher.
Well think about this, is this really possible what you are saying?
I'll submit either truth is a definition, or it must be derived from something (like mathematics or science). I'll omit things like innate knowledge and messages from God as explanations for how we know the nature of truth as they're a bit sketchy from a scientific perspective.
If it is a definition, then the mathematicians and scientists are also spouting "baseless opinions" along with the philosophers, and trust me, everything the mathematicians and scientists said came out of a philosopher's mouth first, probably hundreds of not thousands of years ago.
If you somehow set out to derive the meaning of truth through mathematics (or science), then to avoid assuming what you're setting out to prove, your derivation must contain no truth.
> Is it? What are some other ways of interpreting results?
Sure. There's about as much basis for a material assumption as an idealistic one. The only difference is whether you're willing to assume, with ability to verify or falsify, the notion that an objective reality exists. This is of course a popular thing to believe, but it's ultimately an article of faith.
> I'll submit either truth is a definition, or it must be derived from something (like mathematics or science). [...] If you somehow set out to derive the meaning of truth through mathematics (or science), then to avoid assuming what you're setting out to prove, your derivation must contain no truth.
As someone who regards himself as a scientist, I'm quite happy to just say, "I don't know how to derive the meaning of truth" and move on. Saying "I don't know" isn't spouting a baseless opinion.
We've got enough of an operating understanding of what truth means to function. You do, too. If I punch you in the face and then tell you I didn't, you'll still be mad at me, because you know the truth. Perhaps neither of us can derive the meaning of truth, but it would seem that for practical purposes, deriving the meaning of truth isn't particularly important.
To be clear, what I'm saying here is that philosophers don't know either. Your argument is basically "other fields aren't experts on this thing, therefore philosophers are". No, they aren't. You've actually come closer to making an argument that the derivation of truth is unknowable than that philosophy knows anything about the derivation of the meaning of truth.
> If it is a definition, then the mathematicians and scientists are also spouting "baseless opinions" along with the philosophers, and trust me, everything the mathematicians and scientists said came out of a philosopher's mouth first, probably hundreds of not thousands of years ago.
That's a purely semantic trick. Thousands of years ago, the fields of mathematics and science weren't developed enough to be separate fields, so you could just as easily call those early thinkers mathematicians and scientists as you could philosophers. The thing is, a lot of ideas from thousands of years ago are just regarded as wrong. Pythagoras identified as a philosopher as much as Plato or Aristotle, but given the Pythagorean theorem has stood the test of time, mathematicians are happy to consider him as one of their own today. But neuroscientists aren't rushing to claim Plato because his idea of the separation of mind and body is flatly wrong. If any of those early thinkers are considered purely philosophers today, it's only because no other fields want them. Philosophy today is the leftover dregs of all the other fields splitting off as they matured.
> Sure. There's about as much basis for a material assumption as an idealistic one. The only difference is whether you're willing to assume, with ability to verify or falsify, the notion that an objective reality exists. This is of course a popular thing to believe, but it's ultimately an article of faith.
There's a basis for the material assumption in usefulness. Materialist beliefs let us reason about the world, and predict things. Going back to a previous example: we can both predict that if I punch you in the face it will hurt. You can't even reasonably argue with me on this, because even you don't disbelieve the materialist assumption with regards to this.
The idealist assumption arrives at no such result. You can claim that a mind exists separately from the body and the only thing that assumption allows you to do is get a philosophy degree.
EDIT: To be clear, if we ever meet in person, I will not punch you in the face. That's meant to be a clear example, not a threat. :)
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.
Absolutely not. The domain of philosophy is precisely anything which cannot be formalized with mathematics. Not only does philosophy not need mathematics, math is utterly useless for any philosophy.
The very question “Does philosophy need mathematics” cannot be answered with mathematics.
...and this is why no one needs to care about philosophy. If philosophers insist that philosophy needs neither empirical evidence nor formal proving, then all that is left is made-up bullshit.
There are plenty of people doing actual, real, non-made-up research that might be called philosophy, but since they have evidence and formal logic to prove their ideas, they generally don't have to call it philosophy.
This attitude defeats the scientific method! There’s a motivating state of inspiration prior to evidence collection. It’s the hunch. Every scientific development begins at a hunch.
More or less, what I’m referring to above, is the generalization of this “hunch”.
This is called a "hypothesis", a concept which science doesn't need philosophy to define.
As I said before, sure, you could call this philosophy, but a philosopher without any practical science experience isn't going to have anything interesting to say about it, so I'd just as soon call it "science" and cut the unqualified opinions out of the conversation.
If motion of matter can be formalized with mathematics, this rules out supernatural processes. How would you derive this conclusion without mathematics? Philosophic conclusions are often broad, so naturally they use all available tools to draw insight from.
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."
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.
Berkeley wasn’t the only one to criticize the foundations calculus. And insofar as his criticism of calculus goes, it is IMO hardly more philosophical rather than mathematical compared to criticism from mathematicians like Rolle. Insofar as the “philosophical” side of his argument goes, the general shtick behind Berkeley’s criticism seems to be the desire to show that the foundations of religion were no worse than those of math (he was a bishop after all). So I don’t see how his philosophical insights could help math to develop.
First, you're right about Berkeley's motivation. Specifically he was rebutting Edmund Halley. But his criticism was correct. And he was also correct that through compounding errors mathematicians arrived at truth, if not science. Furthermore he demonstrated that he had a better grasp of calculus than most who criticized him at the time.
You're also right that what he said could not help math to develop. But that is not due to any shortcoming on his part. Until mathematicians realized that there was a crisis, nothing could have made them pay attention. And once a crisis was realized, what would help them find a path forward depended on the details of said crisis.
Now you pointed to Rolle. Yes, Rolle criticized infinitesmal calculus. But he did so before it was well understood, and all of his criticisms were rebutted. In fact Rolle wound up convinced about calculus, and is known for an important theorem. (That said, his original version was considerably less general than the current one is.)
He was not alone. Lagrange also criticized calculus. His alternative was to found everything on formal power series. But his replacement didn't hold up. In fact Fourier's examples were more of a problem for Lagrange's approach than infinitesmals! So from Lagrange's criticism we got the f'(x) notation, but nothing that could help later.
Lagrange's failure to be able to address the future crisis, even though he recognized the problems that lead to it, demonstrates that Bishop Berkeley never had a hope. Lagrange was one of the top mathematicians of the 1700s. If he could not anticipate what would be needed, how could a bishop be expected to do better?
Math is an agreement to explore properties of a well defined set of objects and their relationships using logic.
Philosophy is when you broaden the set of objects to include the human experience but you are still bound by logic, unlike say, poetry (or everyday chatter) where you can use language to deliberately (or often unwittingly) defy logic for emotional effect.
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.
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 + 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.
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.
The effect of a brick falling on someone’s head is not a philosophical question, and neither is one about the common that clearly exists between, say, three horses and three apples you want treat these horses with.
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.
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.
> 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.
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.