☯ My Attitude Towards the Philosophy of Mathematics

Despite my initial exposure to math was from an entirely empirical and hands-on perspective (namely, the physicists' way), My attitude towards the philosophy of mathematics is more inclined to formalism.
Philosopically, math is a deductive system equipped with a set of axioms and rules of inference. The truth of a mathematical statement is determined by the logical consequence of the axioms and rules, and may not infer a certain meaning in the real world. In this sense, math can be seen as a game of symbols and rules.
However, the choice of the axioms and rules is not arbitrary, but is guided by the simple logic of human beings. The development of math was never based on rigorous logical systems, but was driven by the intuition and "what seems to be true". The axioms and rules are then chosen to be consistent with that intuition. Therefore, mathematicians uses natural language to think and do the reasoning, instead of actually relying on a pure formal system.
Of course, the process brings about many problems, for example, the inductive property of the natural numbers is something that consists with our intuition, since it generally describes the process of unstoppably doing something means that everything can be done, which is a simple extention of our intuition with finite number. And we know today that it is true within the Peano axioms. However, such extention can not be applied simply to uncountable sets, as it's not the result of any axiom (I am not getting into well-ordering and transfinite induction at the moment). Similar case is the axiom of choice, which can be intuitive to some people, while not to the others. The intuition of AC comes from the simple idea that "we can always choose something respectively from a set of non-empty sets", but AC itself is an axiom that one can choose to accept or not. This also points out the nature of math, that is, people choose some, but not all of their simple intuitive logics, formalize them to create an axiom system. Why not choose all of them? Because the natural intuition of human beings can generate contradictions (e.g. Russell's paradox), and we only choose and formalize those that assures (although we can't) consistency. Then, one may also ask, does the axioms have to be intuitive? The answer is not necessarily. People do study for example, ZFC but the axiom of regularity is rejected, which is the basis of a whole branch of set theory called "non-well-founded set theory". One can also reject all existing axioms and create a new set of axioms, or even deduction rules. Then, there's a new math totally different from the existing one. However, it is a question whether such a math is useful, or can effectively model the real world as we hope it to be. As I emphasized before, math is just a game of symbols. It's just that we choose to play the one game among the many that happens to be useful.
The intrinsic complexity of math comes from that in real world, there is no such thing as "infinite" or "uncountable", and the concept of "infinity" is purely mathematical. Even as much as the sands in the universe, it is still finite. Therefore, one's acceptance of a certain axiom is purely based on one's stance on the philosophy of math, and is not a matter of "truth" or "falsehood". In general, we choose to accept "infinity" becasue it is interesting and useful, with which the world of math is way richer, we are able to study more things, and potentially have more applications. With that said, one can also choose to reject "infinity", but then we will have much less useful tools. Such an axiom system may be enough for a computer scientist, but definately not for a physicists.
Of course, concepts of analysis are heavily based on the concept of "infinity", and analysis is everywhere inside and outside of math whenever we want to study continuity. How can someone say that "infinite" does not exist in real life? The is a question I'll get to later.
An Illustrative Example of Why Infinity is not Realistic Based:
One familiar with model theory may know the famous Lowenheim-Skolem theorem, which states that any theory with infinite model has a countable model, a famous "paradox" relating to which is the Skolem's paradox. We may now consider a countable transitive model of ZFC: \(\mathbf{M}\), with a stronger meta theory (say, ZFC + "there exists a strongly inaccessible cardinal"). We have \[ \mathbf{M} \models \mathcal{P}(\mathbb{N}) \text{ is uncountable} \] or more precisely, \[ \mathbf{M} \models \neg (\exists f: \mathbb{N} \to \mathcal{P}(\mathbb{N}) \text{ is a bijection}) \] However, since \(\mathbf{M}\) is countable, meaning that in the meta theory, there exists a bijection between \(\mathbb{N}\) and \(\mathcal{P}(\mathbb{N})\). From the countable condition \[ ZFC + \exists\kappa \vdash \exists f: \mathbb{N} \to \{x\in \mathbf{M}: x\in \mathcal{P}(\mathbb{N})\} \text{ is a bijection} \] Of course, there is no real "pradox", but things goes out of instinct when we are dealing with "infinite" objects. Two observation can be drawn from this type of paradox:
  1. The universe of \(\mathbf{M}\) is not rich enough to contain the bijection between a set \(A\) and \(\mathbb{N}\), therefore a set can be "uncountable" in the universe of \(\mathbf{M}\), while "countable" in the meta theory.
  2. A set \(B\) described by a first-order formula in the universe of \(\mathbf{M}\) may not be the same as the set \(B\) described by the same formula in the meta theory. Since any element \(x\) of \(B\) in \(\mathbf{M}\) will be extraly constrained by \(x\in\mathbf{M}\). In other words, the set of the same formula contains less elements in a universe that is less rich.
Here point 2 it is especially important as "a power set is always larger than the original set" is a theorem in ZFC, and should holds even in the meta theory. In this sense the former example could be confusing until one realizes that \(\mathcal{P}(\mathbb{N})\) is not the same in the universe of \(\mathbf{M}\) as in the meta theory, allowing it to be equally large as \(\mathbb{N}\).

🧲 My Attitude Towards the Relation of Mathematics and Physics, and Other Applied Subjects

However fancy the theories can be, Physics is a subject of experimentation. Math is only a tool, a language to describe the world, and it works because it appears to be. In other words, math builds a model to describe the world, and physicists does refinement to it throughout the history.
The volume of an object is something we can observe, and we build a mathematical tool called "calculus" to describe it, and we are happy to see that it fits well with our observation. When we say "fits our observation", we expect that the concept "volume" is tied to the mathematical defination of integration, and every properties about integration provable from the axioms then becomes a property of the real-world physical volume. If not, we will need to find a new formula to describe volume.
Then, we are interested in how math can develop while agreeing on the elementary properties that the tool is primarily intended for. Starting from "integration" and "volume", we construct measure, which agrees on the elementary part of integration, and allows us to study some less well-behaved functions. This is useful because the limit of a sequence of well-behaved functions is not necessarily well-behaved, and Lebesgue Integral provides a closure of functions under operations we are interested in. In other words, physicists may not be interested in the Lebesgue Integral of those not well-behaved functions, but they can be useful in proving theorems about the well-behaved ones.

With that said, if mathematicians chose another set of axioms, the math would be different, but the physics would not. Then, our description of physics depends on the set of mathematical language we use, and could be a bit different from what we see today. Imagine this scenario: physicists one day discovered that the volume of a specific shape deviates from what the calculus predicts. Even this happens, we can still rightfully say that the math isn't wrong, only that the physics needs to be refined, in this case we will need to consider a new formula that describes the volume of that specific shape. Even further, what could happen is that one day people abandon all existing math, and build a new one using new axioms and even new reasoning rules. Then, we may have a much simpler description of physics and other application subjects. In this sense, the new math can be seen as more useful than the old one, but it is not necessarily "more true". The process is similar to the first time people discover that "the earth goes around the sun instead of the other way around" could be a more aligent theory. From today's view, we can say that the heliocentric theory is more useful, but not necessarily "more true", as neither of them is (lol). But of course, one thing could happen is that we may not use any our natural language to describe or do reasoning with the new math any more, and everyone will have to stick with the rigorous formal system for mathematical research.

The same applies to other applied subjects, such as statistics, which can be seen as an application of probability theory. The probability theory on the other hand, despite its modern measure-theoretic formulation, is studied in other settings as well. For example, people also study probability theory without the countable additivity. Eventually, one can argue that there is no probability in reality at all, only the lack of information. As Einstein once said: \[\text{“God does not play dice with the universe”}\] In spite of that, at the end of the day, we stick with measure-theoretic probability theory because it provides stronger results than the others, and is fitting well with real-world applications as far as we can see.

🌌 Metamathematics and Metatheory: Is the Origin of a Formal System Rigorous?

We use symbols to play the "math game", and we uses metatheories to study the "math game" itself. Math game is rigorous, but what about the metatheory? In my understanding, there're two layers of metatheory:
  1. We define the formal system in a way that every human can accept and write it on a piece of paper. For example, we inductively define what a formula is and what a proof is, so that when given a string, everyone seeing it can determine whether it is a formula or a proof within finite amount of time. This is the first layer of metatheory, and it defines the "rigorousness" of the formal system. So that it is the same for everyone, or can be carried out by a machine.
  2. We use another formal theory as our metatheory to study the object formal theory. This appears along with the first layer of metatheory mentioned above, and is used to study the consistency, completeness, and other properties of the object formal theory. In other words, we want to use mathematical approaches to prove things about the object theory, and the metatheory for this is selected to be the one that does the job best. For example, people uses first order logic + PA as the metatheory to study any formal system containing PA, and that gives us the famous Godel's incompleteness theorem. However, this metatheory is not enough for studying the model of the object theory, for which we may need a stronger metatheory. This is the second layer of metatheory.

The two layers of "meta" are entangled, and are both important in forming rigorous math. For example, when people use PA to study ZFC, using Godel number to encode the formula and proof is a part of the first layer of metatheory, and the proof of Godel's incompleteness theorem using axioms and reasoning systems of PA is a part of the second layer of metatheory.

An interesting thing I've been thinking about is, that one can use a less rigorous metatheory (say, natural language) to study a formal system. That way, you can still prove metatheorems about the object theory using the less rigorous metatheory, but your belief in those metatheorems sherely depends on your belief in the less rigorous metatheory. For example, one may philosopically believe the logical reasoning made by natural language are valid, then they can believe the metatheorems they proved using natural language is correct. Believing in the simple logic described by natural language is like having faith in a religion, and may somehow be a topic of philosophy and even psychology.