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.
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.