Fire is a good example of matter that human beings discovered; Fire has been a part of nature even before human beings found it and at some points and we have used it ever since we discovered the existence of fire.

Then, what about numbers?

Is 0 invented or discovered?

I see some postings that argue that 0 was "there" and we found it, and some say that there had been no concept of zero before we invented it.

However, for number pi, area of the circle has not been changed ever since creation of the universe. Then, is pi discovered?

Any opinions?


It is not the case that for every clear cut question there is a 'good' answer: was walking on two legs discovered or invented?

Just as the perception of straight lines, edges or corners, small numbers, say, up to 5, seem to be hardwired. Practical experience has given humans a 'feeling' what is to add/subtract, that is, for example, to get from 4 to 5 or to 3. Neither form nor content was first (just like in the chicken-or-egg problem).

Measuring is a refinement of counting and Pi is the name of the attempt to measure the circumference of a circle with its diameter; it can be expressed in infinitely many ways, just as any other number.

The alternative seems more meaningful when considering more advanced mathematics: many 'inventions' from one domain are 'discovered' elsewhere (eg the Gamma function). Lee Smolin in his Singular Universe exposes a view according to which 'mathematics is evoked', so it is neither discovered nor invented and also it is not arbitrarily phantasized.

A more sociological-historical approach would propose that mathematical objects are a specific kind of 'collective representations'. Actual people live in a society which maintains a tradition. In early times observations and thoughts are anonymously 'sedimented'. Mythological figures fullfill the expection about heroic figures, a person who has been the first inventor or discoverer. Unstated contexts and rules offer for late-comers ready-to-use quasi-objects amenable to further perfecting.


Heidegger wrote in his lectures on Metaphysics:

The extent that humans are not at home in their essence is betrayed by the opinion that human beings cherish of themselves as those who have invented and could have invented language, understanding, poetry and building.

Possibly, one can add numbers to this list...

And then, given the dichotomy between discovery and invention; it suggests that Heidegger tilts towards discovery rather than invention, in its inception ...


I would contend that numbers are discovered and not invented. We did not discover the concept of zero until late in human history.

If numbers are an invention, then we could improve them. However, we cannot improve or destroy numbers as they exist regardless of our existence. Thus, they must be discovered.

Descartes essentially argued the exact same thing after saying "Cogito ergo sum."

  • 1
    As a bit of feedback: I tend to agree with you, but there are some obvious counter-arguments: arguably, the numbers in some sense have been improved - the real numbers are the completion of the rationals, the complex numbers allow us to do a lot of things the reals can't, etc. The argument "If numbers are an invention, then we could improve them. We cannot improve them, therefore they must be discovered" is begging the question. Also, most people are probably not aware of what Descartes said regarding numbers after "Cogito ergo sum," so a citation with what he said would be helpful. Feb 19 '16 at 22:45

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