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Short version:

Considering that science is inevitably dependent on mathematics and metaphysics (Kant tried to raise metaphysics to the status of a science, which I find mandatory to improve the quality of scientific knowledge), and mathematics and metaphysics seem to fit better as matters of philosophy, what is the nature of mathematics? Some say it is an art, but that seems utterly wrong (science would depend on art). I ask this to understand what was Kant's path following in his attempt to give metaphysics an equivalent status of mathematics, how metaphysics would be equivalent of mathematics, and what class would mathematics (and so, metaphysics) belong to.

Long version:

Science is essentially the description of natural phenomena. That is, the description of what we perceive with our senses. So, as it is said commonly, science targets empirical truth (verifiable by experience, although not necessary and universal, like the earth being flat), not final truth (necessary and universal: there's no number which 1 can't be added to). The use of the scientific method is usually what allows knowledge to be qualified as scientific.

Final truths (even if they are unreachable in multiple cases) are the goal of philosophy. Remark that philosophy is said to be the mother of all sciences.

Mathematics seems to fit better on the last category: some part of philosophy. Logical and mathematical truths are necessary and universal (the Kantian definition of pure: non-empirical). Following the same logic, this question essentially shows that mathematics is NOT a science: Is Mathematics considered a science?

But the answer to such question seems just wrong: mathematics would be an art. What??? does it mean that all physical sciences depend on an art? Is art part of philosophy, and science a subset of art? This consideration is not acceptable, for any common definition of art.

The best definition of art that I know belongs to Mario Bunge: any branch of knowledge has three parts: science (the theoretical framework related to the discipline), technique (the application of science) and art (the social application of technique to fulfill some need, either emotional -"this song makes me cry"-, referring to the art or the artist, or functional -"this shoemaker is an artist, he makes the best shoes"-, -"making solid buildings is an art"-). Other definitions of art are trivial or superficial, mainly pointing to esthetics or ideals; in no case art is related to logic or the definition/search of truths.

So, mathematics cannot be an art. It clearly fits into philosophy, and it clearly makes most sciences dependent on it.

Another approach to the same problem is Kant's quest for making a science out of metaphysics. It seems quite clear to me that it is mandatory to define the axiomatic foundations of a metaphysical framework upon which further scientific knowledge would be developed. But Kant seems to have had the same problem: where to fit metaphysics? So, he accepted for his metaphysics to be considered a science. But due to translation and linguistic issues, metaphysics cannot be considered a science nowadays. Part of metaphysics should be at the same level of mathematics, science being dependent on both.

So, what exactly would be the role of mathematics in philosophy? What is the category it belongs to? What is the nature and class of mathematics as a branch of philosophical knowledge?

With this answers, I expect to understand better Kant's project and method to raise metaphysics to a higher philosophical status, perhaps equivalent to mathematics.

UPDATE-2021/07/23: A key attribute of scientific knowledge is testability, empiric observation and prediction. Mathematics being considered a 'formal science' implies that the principles of mathematics can be empirically tested, and that's evidently wrong. That's why it seems bizarre accepting maths to be a science. If you propose considering math a 'formal science', I have no problem accepting that mathematics is a science (yes, a formal one), please just provide an acceptable definition of science clarifying how the idea of a "mathematics science" fits empiric observations/predictions and testability.

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    Prior to the 20th century, the word "science" was used much more broadly, than it is today. Back then, it could refer to non-empirical domains of knowledge. Logic and mathematics were called sciences under this usage. When people talk about making metaphysics a science, they are relying on this older usage.
    – David Gudeman
    Commented Jul 17, 2021 at 21:50
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    Kant's project was to eliminate metaphysics, as then understood, not give it the status of mathematics. He explains why that would be impossible, mathematics only deals with the form of things, hence is susceptible to a priori construction, metaphysics tried to deal with substance. And you seem to operate on the idea that everything must be philosophy, art or empirical science. But modern classifications distinguish empirical and formal sciences that mathematics is one of. Metaphysics, on the other hand, is more of a speculative art.
    – Conifold
    Commented Jul 18, 2021 at 21:24
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    "Final truths are the goal of philosophy". That smuggles in an entire infrastructure around what truth is and how it works. There is no final anything. In a gigantic space of the variation of the fundamental physical constants, 'universal' itself is highly questionable. Truth is always contextual, and relative to a system of evaluating. Even mathematics, which fundamentally is a system of symmetries & recipes to move between them.
    – CriglCragl
    Commented Jul 22, 2021 at 9:32
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    Very related, not identical/redundant: philosophy.stackexchange.com/questions/47586/… , is math a discovery or invention
    – Al Brown
    Commented Jul 23, 2021 at 0:09
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    Kant had the idea that certain types of empirical inquiry (classical mechanics, for example) fall under categories delimiting their subject matter that can be leveraged into a priori conclusions about the subject. This gives rather minimal "metaphysics" and the idea is long discredited. Skipping its criticism by speculative idealists and subsequent metamorphoses, what it morphed into is speculative construction of fallible frameworks/paradigms that can be used for generating hypotheses and evaluating them on epistemic/pragmatic benefits to make up for underdetermination by testing.
    – Conifold
    Commented Jul 23, 2021 at 7:06

2 Answers 2

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I've long believed that mathematics is "the art of formalizing the processes of intellection."

Its uses in science and philosophy arise, not because maths describes reality, but because it allows one to formalize ones thoughts and deductions about it in such a way as to permit allegorical parallels and algorithmic manipulations.

As Einstein put it: "So far as the propositions of mathematics refer to reality, they are uncertain; and as far as they are certain, they do not refer to reality."

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Sometimes a phenomenon can be described as moving from the realm of philosophy/metaphysics to the realm of mathematics when it has been more-or-less fully understood.  I would like to mention a particular case study that was explored in

Mormann, T.; Katz, M. "Infinitesimals as an issue of neo-Kantian philosophy of science." HOPOS: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236-280. http://doi.org/10.1086/671348 and https://arxiv.org/abs/1304.1027

The neo-Kantian philosopher Hermann Cohen was the leader of the Marburg school (1900 plus/minus a decade or two), that included also such luminaries as Cassirer.  Cohen was particularly interested in infinitesimal calculus.  He sought to introduce a distinction between two types of number/quantity: intensive and extensive.  Infinitesimals would be merely intensive, better-known quantities such as pi would be extensive.  Not many mathematicians were convinced.  Abraham Fraenkel, who at some point was attending classes taught by the neo-Kantians including Cohen, was turned off by some of their writings about infinitesimals.  Ironically, Abraham Robinson was Fraenkel's student.  In Robinson's theory, there is a distinction between standard and nonstandard number that, while not identical to, has some similarities to Cohen's distinction.  As Fraenkel himself put it in his autobiographic work, "Robinson saved the honor of the infinitesimal".  In this case at least, one can claim that infinitesimals, that for centuries had been described as metaphysical anomalies, finally moved into mathematics in 1961:

 Robinson, Abraham.  Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A 64. Indag. Math. 23 (1961), 432–440.

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  • upping your game +1
    – user67675
    Commented Oct 5, 2023 at 11:06
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    @prof_post well he is a professor of mathematics at an eminent university!
    – Kristian Berry
    Commented Oct 8, 2023 at 21:24
  • ha cool @KristianBerry
    – user67675
    Commented Oct 8, 2023 at 21:44
  • yeah i recall that now. sorry if i seem too anomic.
    – user67675
    Commented Oct 8, 2023 at 23:02

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