Is mathematics a science? I have long considered this to be open to debate, but my professor said that he once heard the quote,

"Mathematics is not a science. It is a language 
 that is free of contradictions, used by all sciences." 

It seems to be that most non-mathematicians do not think math is a science, while most mathematicians do.

What do you guys think?

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    $\begingroup$ It depends of your definition of science. $\endgroup$ – A.L Mar 14 '14 at 3:39
  • $\begingroup$ Please define "science" in terms that mathematicians will support (not just comprehend). For most definitions it is then easy. $\endgroup$ – Ross Millikan Mar 14 '14 at 3:41
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    $\begingroup$ I was on a jury once, while still in the choosing phase one lawyer asked me "Mathematics is a science, right?" to which I replied "Alright" sort of drawn out; everybody laughed but him. $\endgroup$ – Will Jagy Mar 14 '14 at 3:44
  • $\begingroup$ I tend to agree with your professor but how do you know mathematics is free of contradictions? You may want to read Godel... $\endgroup$ – Sergio Parreiras Mar 14 '14 at 3:44
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    $\begingroup$ science (n.) The intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment. $\endgroup$ – louie mcconnell Mar 14 '14 at 3:45


It depends on the definition of science. According to Wikipedia, mathematics it's among formal sciences.

Comic from here.


science (n.) The intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment

Therefore mathematics is a science. Mathematics is clearly an intellectual and practical activity. It also explains the behavior of the world. The only possible doubt is the "through observation and experiment" clause. But now this has become a debate centered on topicality

(debate term :D if you did it in h.s.)


It is not the business of mathematics to explain nature or build models for natural phenomena. That is the job of science. But mathematics provides the tools for scientists to build their pet models. Number theorists would not be happy with the approximations $\surd 2=1414/1000, \pi= 314159/100,000$.

  • $\begingroup$ Well, they wouldn't be happy because $355/113=3.1415929$ is better. Also $99/70=1.41429$. $\endgroup$ – Mark Adler Mar 14 '14 at 5:53

Two viewpoints:

(a) The touchstone of science is experiment. The touchstone of mathematics is proof. Statements in science can never be proved to a certainty, but they assert something factual about the real world. Statements in math can be proved logically, but they only make conditional assertions about the real world, never something directly factual. Thus mathematics is not a science.

A quote from Richard Feynman, Lectures on Physics:

Mathematics is not a science from our point of view, in the sense that it is not a natural science. The test of its validity is not experiment.

(b) The root for the word 'science' is the latin 'scientia', which basically means knowledge. Mathematics is certainly a kind of knowledge. Math is so deeply intertwined with physics, and to a lesser but still significant extent with chemistry, biology, and other sciences, that it makes no sense to separate. Thus mathematics is a science.

Supposedly Niels Bohr once said that the opposite of a small truth is a falsehood, but the opposite of a great truth is another truth. Not good for 2-valued logic, but perhaps a good answer to your question.

(Incidentally, Niels Bohr's brother was Harald Bohr, a mathematician of some note.)

  • $\begingroup$ Statements in math can be proved logically - prove Church's thesis logically! $\endgroup$ – user58697 Mar 14 '14 at 6:32
  • $\begingroup$ Church's Thesis, in so far as it's part of math (and not of philosophy) is a definition -- the definition of "computable function". Compare with "continuous function", which has a formal definition inspired by a intuitive notion. Anyway, I was offering one viewpoint, hardly original with me. $\endgroup$ – Michael Weiss Mar 14 '14 at 11:22