When I was an undergraduate, I was a math major and took an unofficial philosophy minor. Why I chose these, and in this order, I can't really remember--maybe because I didn't want to be a pre-med, or was decently good in math, or because a classmate's enthusiasm led me to it.
I never became a mathematician, though I did do some professional computer programming, and more of that years later as a graduate student. I never took further philosophy courses, but kept my interest, both as a graduate student and then during my career as a geneticist and epidemiologist.
Math is of course a tool with many practical uses, and discovering facts about logic and numbers and the like is of interest in its own right (to some people, at least, including those with the level of abilities needed to probe these sorts of things). But mathematics is not just about practicalities of quantitative things. It's also about logic, and reasoning, and mind-bewildering things like aspects of 'infinity'. That's a word easy to write but a concept very difficult to intuit (even if one can, properly trained, manipulate infinities mathematically).
Infinity can be handled mathematically, but what does it mean? How can some thing in or about Nature actually be 'infinite'? We can write down numbers without ever coming to an end, and mathematicians can write down theorems (and even prove them) about infinity--indeed, about levels of infinities!
But what about the real world? Can space, for example, really be infinitely large (which, among other possible interpretations, means it goes on 'forever', without any kinds of boundaries). Well, we can imagine that there is nowhere a Keep Out fence at the limits of space, with literally nothing (not even 'nothing'!) on the other side.....well, can we really imagine that?
In fact (not fiction!), mathematicians have long dealt with different levels or degrees of 'infinity'. There are the integers 1,2,3.... which go on without end, at least in principle. No matter what number might seem to be at the end, you can always add a '1' to it and write the next number in line. It's a countable kind of number even if it's not practically countable.
But infinity can be like the number of points between any two spots on, say, a ruler. Like, say, one inch. That number is not limited. You can identify tenths of an inch, and make little marks, or hundredths, or thousandths.... but this has no end. Of course, you can't actually physically find such spots on a ruler, but they exist in theory.....don't they? How do we know that? How do we know that an inch in some ruler or some place in space, actually has trillionths of an inch inside it? We can write down such numbers, and manipulate them, but how do we know they actually exist?
How can we know that space, our universe, had some sort of 'beginning' and what was before that?
Mathematicians and relevant scientists can write down all sorts of things about these entities.....or facts.....or whatever they really are. They can add and subtract them and express them as angles or distances and so on. But is this kind of thing real in some sense? How can we know? How can we know that there are quadrillionths of an inch (indeed, exactly a quadrillion of them!) not just in this or that inch, but in every inch!? These concepts are imagined, and we think they are, in some sense, real. But are they and in what sense?
Why, in this sense, does mathematics actually work, not just on paper in a class but in the real world out there--indeed, our here and there and everywhere?
What about stuff and processes?
Well, it's more than just numbers. How can we know that stars are actually there (or, rather, that they at least were there when the light we see from them left them)? In what sense do we know that gravity is real? If I drop my pen, it will fall at a certain speed that I can measure. But if I pick it up and drop it again, why am I very sure that the same will happen if I drop it again?
The basis of what we call 'science' is the assumption that the universe is orderly, here and everywhere, and that 'orderly' means that it follows certain rules or is controlled by certain forces, or something of that sort, that are general rather than locally ad hoc. For example, the force of gravity depends on various things, like nearby masses, but given that it is universal. And it weakens in an orderly way universally. And carbon is carbon....everywhere!
How do we....how can we....know such generalizations? Here are some thoughts about this....
In our world, they must be true!
One answer to such a question is a rather fundamental kind of assumption. On earth, since the beginning of what we now call 'science', people have observed the regular predictability of things. Long ago, quantitatively inclined observers noticed that they could develop methods to describe and even predict things about the real world--we call that 'mathematics' and it works to extreme precision.
The universe seems orderly in a universal sense!
What we can see of the universe, through telescopes and space-craft suggests this kind of universal uniformity. We haven't explored it enough to be absolutely sure, of course, but spacecraft seem to behave just as we'd expect if physical universality were true. There are, of course, those who posit other universes, and of course there could be different laws of nature in different universes.....although if they are all made from the same beginning (or infinitely long existence), we would expect them to be the same or, at least, to change in some orderly way. So, to me, multiverse theories don't really bear on the question for our universe.
It's not just math--it's the universe--that's.....universal!
We can do math here, there, or everywhere, since math per se is just the manipulation of symbols according to certain rules that we, ourselves, have posited. In that sense, it's just a kind of fiction and indeed once we've somehow decided on our axioms, rules for logical reasoning, and so on--our 'givens'--everything else follows 'automatically' (that is, for those clever enough to make the deductions properly).
What is perhaps most interesting about math is that, given these basic assumptions, mathematical reasoning and the deductions it leads to seem to fit the world....perfectly and universally! There is, to take a baby example, no place anywhere in which 2 + 2 does not equal 4. We might say, indeed, that this equation 2+2=4 is just a description, in terms we devised, of reality. But to assume that is a correct assessment, we have further to assume that the reasoning in that equation is universal. In fact, the symbols and basic precepts are in a sense just descriptions or labels. Mathematicians get very intricate with this, but it boils down to manipulating symbols according to some axioms, or rules, that we assume are universal. Indeed, we define '2' and '+' and '=' in a way that makes the equation a shorthand for what we observe. Complex equations just express more complex situations, and to show that an equation or generalization is 'true' we use rules of reasoning to construct proofs.
It so happens that this was done with the real world in mind, so the results generally apply to the real world, but there could be types of 'math' that have assumptions that don't fit the observed world, and we could see where that leads....just for interest.
But for aspects of math, or other sciences, that we find to apply generally here on earth, or seem to apply to what we can see in space, can we say that they are 'universal'? There may be 'universes' where these things don't work, but these are, so far, just imaginary.
The mystery of mathematics, like much else in science, is the philosophical one: we can make observations, here and there, and then make generalizations about them that we often call 'laws' of Nature, because they seem to apply everywhere, even to areas of observation that had nothing to do with the formation of these 'laws'. Indeed, in the history of science, sometimes data did not do that, and we had to reformulate our theory of knowledge. The obvious case is the way in which observational and theoretical science based on observation replaced received explanations (such as evolutionary biology having replaced biblical Genesis accounts of the living world).
Nonetheless....
Still, science provides what seem to be generalizations but we can rarely (if ever) prove that the latter are universally true, or permanently true....unless just by definition, and that may not be very interesting or even helpful in our wish to understand existence!
It is the philosophy of science that in a sense accounts for our willingness to accept that what we call 'science' is more than just a list of what we've observed so far, and indeed that it applies to what we will see in the future. Careless or casual or philosophically uninterested scientists may just putter along, applying current theories and the like and doing real science in their daily lives. But history shows that at various times, such routine 'normal science' runs into technical, or even conceptual barriers to further progress. At such points we struggle to explain what we see in various ways--as mistakes, for example--or to shoe-horn the findings into the current theory.
We can always account for what we see by saying, for example, that "God made it so." Such an assertion cannot be falsified, but is so ad hoc that it provides no help for what we might want or expect to observe tomorrow. That is where science comes in, because it generalizes about Nature and makes predictions possible and makes new findings fit in with existing explanations.
Indeed, when this is not possible it forces us to seek better explanations, or new theories! If we can't rely on saying "God just made it so," we have to try to fit diverse observations into the same explanatory framework, the same kinds of causes we know about, and so on. That's basically what science is.
The philosophy of science is the study of how and why this works. Many scientists never studied the philosophy of science and scoff at its importance. But they use it, implicitly, every day. That is what they're doing when they do something different from what's been done in the past, but assume it will work because Nature has a structure--it is generalizable. It has regularities--'laws'.
Without belief that Nature really is law-like in this way, science as we know it would be impossible.
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