Is mathematics a science or a language?
The question, of course, follows on the heels of a recent Twitter discussion based on non-compliance with the Popper falsifiability criteria. I am interested in your opinion, I will formulate my own later.
Is mathematics a science or a language?
I have already asked this question here and in general I have worked out some answer for myself. However, in a nearby thread, the issue came up again. They say that there is no experiment, so it does not meet the criteria. Let's discuss this with some links that are a little fresher than Popper.
8 Answers
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I don't see the need to choose either-or.
Mathematics cannot be defined through the subject matter of its study or through the source of its new ideas and concepts. If you take algebra out of mathematics, it will still be mathematics (disfigured, incomplete). If you take out of mathematics the ideas that came to it from physics, it is still mathematics (poor and pale). But if we take the proof out of mathematics, all we're left with are pictures and fiction.
There are experiments in mathematics, of course, but they do not play the same role as in the natural sciences.
In mathematics, there is a tradition: in published papers, experiments are usually omitted, leaving in the shadow those paths that led to the result. The public is presented only with proofs, which sometimes gives the impression that proofs and logical reasoning are the main method of mathematics. This impression does not fully describe the state of affairs.
As in the natural sciences, in mathematics, experiment is a source of new knowledge, ideas, and hypotheses.
However, in physics, an experiment is also a criterion of truth. To test the hypothesis, the physicist will develop a new experiment. In mathematics, the criterion of truth is not an experiment, but a proof.
A trillion experiments to test the Riemann hypothesis do not serve as a proof in mathematics, it is still a hypothesis.
One might say that mathematics is the only science in which proof is the criterion of truth. However, the concept of proof is fluid. The history of mathematics tells us that the idea of what a proof is has changed over time, and it is constantly changing now. There is no generally accepted notion of what is proof and what is not.
There are a lot of books about what mathematics is, some of them are directly called that. I consider the best book on this topic to be Stephen Kranz's “The Changing Nature of Mathematical Proof”, which I once translated.
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Math is a science because
Mathematics originally studied numbers, but now it studies any objects to which calculations can be applied, i.e. rules for obtaining the result of some action based on the original data.
Mathematics studies what actions can and cannot be done and answers the question of why it is necessary to use such actions in order to get the right result.
Numbers were not invented, they are being studied, which means that the laws that apply in the objective world of numbers are being discovered.
Math is not a language because
Any sign, including a mathematical one, is only the name of some objective reality. The name that is used to denote it reveals its essence.
Numbers and other mathematical signs are only the form of the name that is used to represent this name in the form of a sign that is both an image of the name and the one called (they are inextricably linked in the sign).
Any proof in mathematics begins with the definition of signs, which are then used in the proof, and the sign in mathematics always denotes a certain amount obtained according to certain rules.
The subject of mathematics is rules, not signs, and signs can be anything, yes, in mathematics some signs are generally recognized, but together these signs are only a tool for writing down laws, even in mathematics itself.
Mathematics has a language that is a subset of natural language, but mathematics itself is not a language, even in the sense of mathematics being the language of science.
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It would be good to understand what science is. We “find” experimental data and, in general, experimental data of mathematics in our own head. This is how we “construct” reality. But the construction is not arbitrary. Like in poetry, for example.
It is impossible to compare mathematics with physics or chemistry. Mathematics does not have its own substratum. It counts everything and nothing in particular.
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Half-jokingly, but. How is it that there was no experiment? Let's take Fermat's theorem. Substitute specific numbers for variables. Let's perform the calculation. We get the result and compare it with the predictions of the theory. What's not an experiment? Moreover, what is any unsolved mathematical problem but a hypothesis based on empirical observation?
The subject of mathematics is the world of mathematical objects. This is a world that is special in that its objects derived from the initial axioms are what they are and cannot be otherwise (if the axiomatics are not changed). This is a world of formal systems that obviously have special properties. It is all the more surprising that some of its objects and their properties can be used to describe the structures of the material world. But precisely because some are objects that are not connected to the real world in any way, are autonomous from it, and yet have their own existence. If they are “words” of a certain “language”, then we have not yet found anything in our world that can be called them. And it would be too bold to say that we will definitely find it.
So science, of course.
But doesn't physics provide the language of chemistry, and chemistry the language of biology?
Without references, because I don't know which of my theses needs to be based on authority.
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Math is part of physics. Physics is an experimental, natural science, part of natural science. Mathematics is the part of physics where experiments are cheap. (V. I. Arnold )
Recent post on Yandex Zen :-
Quantum field theory – quantum electrodynamics.What is it?
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Mikhail Kolonutov comments
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QED = Quantum Electrodynamics
QED is nothing more than a language for describing phenomena, but it is not a way to understand the physics of processes occurring.Follow Feynman's rules and you will get the desired result, but do not ask why this is so.
It is enough to believe that a cloud of virtual photons constantly follows the electron, surrounding it with energy quanta.It is enough to believe that electrons repel each other because they aim carefully and throw a virtual photon at each other.
And how does QED as a physical science differ from religion, which also requires blind faith in the destinies of the great?
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Commentator doesn't know :-
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According to one of the leading experts in superstring theory, Brian Green, Maxim Kontsevich brought this theory out of a dead end with his work. Kontsevich gave a mathematically rigorous formulation of Feynman integrals for topological string theory through the concept of the module space of stable maps introduced by him. Knot theory, closely related to attempts to combine superstring theory with general relativity, is also the domain of Kontsevich's successful work.
Kontsevich integral, a topological invariant of nodes (and links) defined by complex integrals analogous to Feynman integrals,and generalizing the classical Gaussian “connection number”.
Смотри https://wikichi.ru/wiki/Linking_number
In topological field theory, he introduced the module space of stable maps , which can be considered as a mathematically rigorous formulation of the Feynman integral for topological string theory.
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The basics of mathematics are endowed with the function of language. Numbers – letters, mathematical signs-punctuation marks. Without knowing the basics, all the beautiful formulas and sub-and superscript icons turn into garbage. Knowing the basics of the language turns mathematics into a science.
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This can be interpreted either way, depending on what the goal is.
Of course, mathematics is a science, because it meets all the criteria of science. The experiment is not a necessary condition for this. For example, there are no experiments in history or geography either, right, but it is unlikely that anyone will doubt that these are sciences? In mathematics, by the way, the experiment is replaced by some tentative assumptions, which are then justified or refuted. For example, it is typical in proofs to the contrary. An assumption in this kind of evidence is a kind of experiment.
On the other hand, mathematics can also be interpreted as a language, if you are interested in this particular aspect.
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I will express myself very correctly.
It seems that this is a fad on Q among explainers-to answer questions, forgetting about definitions.
Is mathematics a science at all? Not in the philistine sense as something to be studied and applied, but in the epistemological aspect? Accounting, for example , is certainly a system of knowledge, very useful and demanded, and quite complex, but it is by no means “a special type of human cognitive activity aimed at obtaining, justifying and systematizing objective knowledge about the world, a person, society and knowledge itself, on the basis of which a person transforms reality” (the first definition of science that comes to hand).
And what is mathematics with this view? Yes, she is a servant of science, of all sciences, a jack-of-all-trades, but is she science itself? If strictly?
Write “Of course, mathematics is a science, because…” and “hardly anyone will doubt” – this is too little.
Doubting is useful. Without a doubt, there would be no science itself.
But it is contraindicated for an accountant to doubt the essence of his craft. Decide on this for yourself.
Expert citizens, I do not insist on anything, but if you are going to explain something, explain it from the beginning. You should not rely on intuitive understanding and imaginary evidence.
I don't have a Twitter account, so your “of course” doesn't mean anything to me. But it says “non-compliance with the falsifiability criteria”. Here is a live example of meeting these criteria. Fermat's theorem says that a well-known equation has no integer roots for n greater than 2. Find such a solution and voila – you have falsified the theorem. It has already been proved, so we know for sure that no such roots can be found, but we can take any unsolved problem instead – it is unsolved because it is not refuted. If it had been refuted, there would have been no problem.
Mathematics is both a science and a language.
Mathematics as a science discovers and predicts new knowledge that has not yet been discovered by other sciences.
Mathematics as a language provides a tool for describing any ideas and concepts.
David, for me personally, Mathematics is a means of describing an ideal world. Taking into account the errors of the measuring equipment, which are also mathematically reflected in the description of the real world given to us as a whole, we can say (yes, what can we say!!!) that Mathematics is the language of Physics (as a natural science).
And the concept of scientific experiment given by Galileo is hardly applicable in Mathematics, because if there is a theorem, and it was proved by one person, then another can only reproduce this proof.
Therefore, we cannot apply the concept of reproducibility of an experiment to the theorem, regardless of the location and operator. The predicted result? So either there is a proof or there isn't one. Isn't that right?