What is the reason for saying that mathematics does not constitute science even if use experimental approach so often in mathematics?

Question

Modern Science is divided into three major branches:

$1.$ Natural Sciences (Includes Physics, Chemistry and Biology) Which studies nature in its broadest sense.

$2.$ Social Sciences (Includes Economics, Sociology and Biology) Which study Individual and societies.

$3.$ Formal Sciences (Includes logic, Mathematics and Theoretical computer science) Which study abstract concepts.

There is disagreement, however on whether the formal sciences actually constitute a science as they do not rely on empirical evidence (i.e. Information received by means of senses, particularly by observation and documentation of patterns and behaviour through experimentation)

My question is what is the reason for saying that mathematics does not constitute science even if use experimental approach so often in mathematics also?

In words of Paul Halmos When you try a theorem, you don’t just list the hypothesis and then start to reason. What you do is trial and error, experimentation and guesswork. You want to find out what the facts are and what you do is in that respect similar to what a laboratory technician does. In fact there are so many evidences that mathematics is as experimental as other sciences. For instance consider

$\color{purple}{n^2+n+41\ is \ a \ prime \ number\ for \ all \ natural \ numbers}$.

We check this for natural numbers and find out that it is true for numbers $1-39$ but fails at n=40 so this result is not true. Same way we see that $\color{navy}{F_n=2^{2^n}+1 \ is \ prime \ for \ n=1,2,3,4}$ but not afterwards.

One reason I can think is experimental mathematics is relatively newer and we haven’t accepted its importance .

Other reason can be that in other sciences you take some observations to support, refute or validate a hypothesis but in Mathematics your result might come out to be false after so many positive observations or sometimes you cannot prove/get close to validity of result even after so many observations. Like in given examples:

$\color{brown}{a.} \ \ (n^{17}+9,(n+1)^{17}+9)=1$

(First counter example is $n=84244329255928893292881973223089006724459420460792433$) A physicist would be convinced by experimenting even less than $33$ times :)

$\color{blue}{b.}$ Smallest value of $n$ for which $f(n)=991n^2+1$ is perfect square is $n=12055735790331359447442538767$

$\color{red}{c.} \ \ 1000099n^2+1$ is perfect square for first number having $1116$ digits

I might be wrong in some places so please mention if you find any.

References: Wikipedia articles ( Experimental mathematics, Science)

Answer 1

While experiments play an important role in mathematics, it's not the same as the role they play in the experimental sciences. In the latter experimentation is truly indispensable; in the former, it is instead a source of inspiration, with the actual "finished product" being a mathematical proof which could in principle have been discovered "ex nihilo."

Of course, mathematics isn't actually done without experimentation in practice. This is especially true if we construe the word "experiment" to include proof attempts, which I would argue we should: often the intuition for a proof of one theorem comes from the difficulties one hits in trying to prove the opposite. But this is a fact about practice as opposed to nature; a computer can search blindly for a proof of a given sentence and the validity of such a proof (if discovered after all) is independent of the fact that there was no experimentation employed in its discovery. (Ignore for a moment the fact that this blind proof search is infeasibly time-consuming; again, that's not the point here.)

So the question comes down to this:

When we declare something an "experimental science," what role must experimentation play in it?

I - and I think many others - would argue that it's not enough for experimentation to be important in practice; rather, to genuinely be an experimental science, a subject must take experimentation to be of central importance. If we accept this, then mathematics is not an experimental science since the centrally important objects are mathematical proofs rather than mathematical experiments.

Answer 2

This is more of a question in philosophy and semantics. The question of whether or not we define mathematics as a science is not a functionally useful question by itself. Its usefulness arises when we compare and contrast how math and science are related in their methodology and application.

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Demarcation

For starters, demarcation of the sciences is a fundamental problem in the philosophy of science, but there is no such analogue in mathematics. There is not a singular answer of what makes something considered "science". Without getting too deep into specifics, there is a gradient of what is considered psuedoscience versus science, which is generally the same metric we use to label "hard sciences" versus "soft sciences", very similar to the differentiation you describe as "natural" versus "social" sciences.

On the other hand, classifying something as math is more straightforward. The only relevant question is whether or not the statement is well-founded under some axiomatic system. (My wording here may not be technical enough, feel free to correct me.)

Methodology

While math can rely on experimental computations as evidence, they never constitute a real proof. Math can rely on hard logical proofs that (under a certain axiomatic system) can be guaranteed to never be wrong (assuming the proof is constructed correctly). We can demonstrate with absolute certainty that there are infinite prime numbers.

Like you mentioned before, science relies on empirical evidence, so we can never use a purely logical proof for anything scientific. We apply mathematical models to our scientific theories because the empirical evidence lines up with it, but we can never be sure that there aren't any exceptions that break under that model. Consider classical mechanics vs quantum mechanics. We used to believe that we can just add velocities linearly, because under every empirical measure it was close enough to being correct. If we treat a set of empirical data as a set of discrete points under some function, there will always be an infinite (and uncountable) number of functions that satisfy those points. Science aims to find the simplest function that models the data and corrects for accuracy over time as new information arises.

To make matters even more difficult, science is rarely (possibly never) a closed system. We can idealize closed systems in order to attempt to understand something about the system, just as how we apply mathematical models to our data, but almost no scientific finding occurs in a truly closed system. Our empirical measurements are always prone to be altered by something outside of the context of our findings. This is not the case in mathematics. We may see an overlap of different fields of math, but it is always self-contained within the axioms, which form a closed system.

Application

Because of the reasons listed above, scientific discoveries are structurally different in application. As findings are produced, we need to update the meta data and conclusions accordingly. This could mean that established findings could be refuted and overturned within the community. Findings that reference those findings would then need to be revised or looked over again. This doesn't really happen much in math unless there is an objective logical error found within a paper. Even the cases where something similar to this happens, the rigor of modern mathematics makes this less and less likely, and often times it could just be working under a different context, which is valid as long as it is self-contained.

Most all sciences are built upon the formal sciences, or math. No matter what field, all results and conclusions need to be founded logically, and the nature of this logic's existence is just accepted, despite its intangibility. There is a debate about whether math was discovered or invented. The school of thought that math objects just inherently exist is called Platonism. The school of thought that math objects are the result of human-invented rules is formalism. If you accept the Platonism position, you could argue that math is a science, as these are observations on existing objects. If you accept the formalism position, then this contradicts the idea that we are studying the natural world, but rather manmade constructs and ideas.

Epistemology

Humans are inherently inductive learners. Learning is a process that requires a balance between hard memorization of previous outcomes and abstracting concepts and generalizations from them. For instance, if you touch a hot surface and feel pain, inductive reasoning would lead you to believe that you would not want to touch other similar hot surfaces. There is no logical reason to assume that other hot surfaces would similarly cause you pain outside of induction. You could argue that scientific studies can help explain the concepts of heat, touch, and pain, but remember that scientific reasoning is also inductive by its definition.

While math is largely a deductive process, humans gather inspiration about how to engage in the process through inductive reasoning. We can find a conjecture that we attempt to approach by some inductive process until we find the deductive proof. In this way, both science and math are the same. You could argue that any field that ever existed similarly requires inductive reasoning, but the goal of science and math is to formalize the inductive process into something more measurable.

Answer 3

Science is falsifiable. You being with a statement and look for means by which to disprove it. Advances in science change the information to better reflect the empirical data by slowly overturning the existing model. It's temporal and mutable.

By comparison mathematics is immortal. Once a theorem is proved it's proved forever. There is no way to falsify it because the proof ensure it's always true with the given axioms. The proofs Euclid constructed over a thousand years ago is equally valid today. Nothing will ever change that. While experiments can motivate a conjecture they don't actually prove anything and so no mathematics has been done.

Answer 4

Asking whether mathematics should be called a science is the same kind of question as asking whether a plant should be called a living thing. In other words, it is purely a matter of either convention or opinion, and hence does not belong on Math SE.

However, there is a specific mathematics-based sense in which mathematics is very clearly distinguished from empirical sciences. Every empirical experiment in science only gives you a finite amount of data that you hope will support a hypothesis about a typically infinite domain. We can do so because we make some helpful assumptions, such as that the true relationship is continuous almost everywhere, so that our data is actually representative of the part of the domain that we can 'reach' via our experiments. We also try to find the simplest explanation that fits the data. But in many cases we can never prove our hypothesis right, due not only to experimental error but also the fact that we cannot exclude some pathological possibilities (such as that the universe conspires to feed us misleading experimental data).

In very strong contrast, mathematics revolves around finding rigorous proofs of theorems in a chosen foundational system, or guessing true statements about a chosen structure such as the natural numbers. Once a proof is found and checked, especially if formally verified, there can be no doubt that the statement is logically necessary given the chosen foundational assumptions. Your kind of large counter-examples to mathematical conjectures are really very different from counter-examples to empirically testable scientific hypotheses, because those conjectures are discrete and there is no expectation of any 'continuity', which should also explain why such pathological counter-examples do not show up in empirical science because every hypothesis can only be tested within a certain range, and it is usually about a continuous phenomenon. One is not justified in claiming empirical support for a hypothesis beyond the range 'covered' by the experiments.

Nevertheless, there are a number of examples of 'empirical' experiments in mathematics, even though this is not generally considered to be mathematical results, but merely 'evidence' for conjectures. For example, the k-tuple conjecture is widely considered to be very likely true, based on both empirical evidence as well as various heuristics. Obviously, no empirical test can possibly check the limit of asymptotic density of anything at all, but that does not stop people from believing that the numbers 'look like expected if the conjecture is true', which lends a weak kind of evidence to the conjecture, in much the same way as some statistical tests lend evidence to not rejecting the null hypothesis. Similarly, people believe that $π$ is a normal number, based on statistical tests, but it remains a conjecture.

Also, it may be that more and more mathematical theorems are found that are like the 4-colour theorem, in requiring significant computational effort to prove, which can be considered to be empirical in some sense, though without the experimental errors that come with any empirical scientific experiments.