# Are there well-known mathematicians who shared Arnold's view about mathematics as natural science?

V. I. Arnold asserted that mathematics is a natural science:

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

V.I. Arnold: "On teaching mathematics" (1997)

1. This opinion is rare. Many mathematicians will oppose, in particular pure mathematicians. But is it solitary?

2. Are there famous mathematicians who have shared in written form Arnold's view that mathematics is a natural science?

• This opinion is not rare. It was common until the beginning of 19th century, for example it was explicitly expressed by Fourier. And many mathematicians still hold it. Jun 12 '17 at 14:36
• @Alexandre Eremenko: Can you provide quotes? Of course most interesting would be contemporary mathematicians. When I have seen discussions about Arnold's provocative statement, I have never met a supporter (except myself).
– Otto
Jun 12 '17 at 16:39
• As I am also a supporter, you already have $\geq 2$, besides Arnold. Now, most mathematicians do not make such general statements in print, because they are busy proving theorems. So next time I see this expressed in print, I will inform you. Meanwhile you may read the Introduction to Fourier's Analytic Theory of Heat. Jun 12 '17 at 18:42
• I am not sure if you are looking for the uses of this specific language, or sympathy to the view. Many would say that there are obvious similarities and dissimilarities between mathematics and natural sciences, so this is about drawing an imaginary line. Here is Magidin putting them on the same side of it. See also the 1990-s debate over the Glimme-Jaffe's manifesto on mathematics vs physics, with many big names participating. Jun 12 '17 at 23:49
• I just ran across this old post. I quite agree with Arnold's viewpoint myself. I expressed this in my answer on MathOverflow where I asserted my opinion that just as "physics is at its root a laboratory science where good results must accord with physical reality", so "mathematics at its root is a laboratory science where good results must accord with logic". I would go even further to say that the key role played by experimental design in the natural scie May 5 '20 at 13:02

Newton is a dubious example here, but it is well known that in the preface of his Principia he asserted (p.xiii):

geometry is a founded on mechanical practice and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.

Leaving aside the etymology harking back to 'measuring', for Newton geometry is mathematics, as he disliked cartesianism and its algebraic approach.

Before the XXth.c most of mathematics has been done by natural scientists-physicists; Imre Lakatos, who was a philosopher, proposed in his Proofs and refutations the view that mathematics is just like any other science; later Quinn and Jaffe outlined the idea that mathematics is theoretical and makes hypotheses, for which proofs are offered and so it functions as a natural science.

An explicit statement has been given by Kronecker:

Mathematics is a natural science – not better, not more complete, and not simpler the phenomena can be described than mathematically.

"About the notion of number in mathematics", Public lecture in summer semester 1891 at Berlin – Kronecker's last lecture. "Sur le concept de nombre en mathematique" Retranscrit et commenté par Jacqueline Boniface et Norbert Schappacher: Revue d'histoire des mathématiques 7 (2001).

Gregory Chaitin openly holds this position (though he might dispute the "experiment is cheap" part). He believes that the incompleteness results of Gödel, Turing and himself show that the mathematical knowledge has limits, and (according to him) that the limits are quite near. From that observation, he constructs an interesting argument (interesting in more meanings than one).

A Newtonian physicist, Chaitin says, would, after experimenting with the electro-magnetic field, be led to phenomena that cannot be explained through Newtonian mechanisms and eventually to Maxwell's equations, that is to say to new fundamental principles of physics that cannot be reduced to Newtonian physics. Likewise, the experimental investigation of the microscopic world would led a physicist who has mastered Newton's and and Maxwell's contributions to quantum mechanics and ultimately Schrödinger's equation; another principle that cannot be reduced to former principles.

Chaitin believes that mathematicians should do the same thing. When faced with a problem they cannot solve or with a new phenomenon, they should adopt new axioms, see where these new axioms lead them experimentally and advance in this way, backpedaling if necessary if one axiom turns out to lead to a contradiction. He is perfectly clear that these new axioms should absolutely not be axioms in the Euclidean sense, that is to say evident truths or foundational principles. They may be, indeed should be in his mind, complex, non-obvious propositions, like Schrödinger's equation. As an explicit example of a new axiom he recommend mathematicians adopt and whose consequences he recommend they experimentally explore, he mentions the Riemann hypothesis.

Chaitin doesn't deny that proofs are important. "It's nice if you can prove [a conjecture]" he writes "especially if the proof is short. And if you have different proofs from different viewpoints, that's very good. But sometimes you can't find a proof and you can't wait for someone else to find a proof". In that case, "[mathematics] should be pursued more in the spirit of experimental science, and [mathematicians] should be willing to adopt new principles".

Reference: Randomness in Arithmetic and The Decline and Fall of Reductionism in Pure Mathematics. Gregory Chaitin (1993).

George Pólya gave a similar statement (see here):

“Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper."

The same general point of view is developed in his two-volume book Mathematics and Plausible Reasoning (suggested read if you're interested in this point of view on Mathematics).

Tegmark is a present-day example. You can't find them more Platonic (with a big difference though so maybe he's exactly the opposite). Let's not hope his lovelife is like his view on reality.

He states that the universe is math. All that there is to discover are mathematical formulae. Contrary to Plato though he thinks we stare the mathematical formulae and structures right in the face. They don't merely throw shadows in the material world we live in, like Plato thought. The material world is the world of math and can be known.

Max Tegmark even thinks that there are multitudes of different universes each showing different mathematical systems which can't be compared or translated among each other. A person living in our universe, being a math structure himself, will never be able to understand the other kinds of math.

This view is an extreme one and obviously overlooks many aspects of reality. But if he is happy with it... It is a legitimate view.

David Deutsch is a similarily minded physicist. Also he thinks the are four levels of parallel universes whose Nature is math and that in different universes different mathematics can be encountered. See this article.