I know that many people believe that math is discovered, but here I want to know if pure mathematics, in specific, is discovered or invented and why. There are definitely many arguments to both sides. Personally, I believe that pure math is discovered but my mind could be changed. I know that pure math has gone a bit too abstract recently, but I don’t believe that it makes it any less a discovery. When something is invented there is possibility that the results could have been different depending on the organism that invented it. I don’t think that this applies to math because any organism anywhere would come to the same conclusions. I would love to hear a second opinion.

  • I was trying to get an explanation about pure mathematics soecifically rather than applied because it cannot stand alone as its own science. Thanks though. Jun 20, 2020 at 5:45
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    Most of those who believe that mathematics is discovered refer to "pure" mathematics anyway. Applied mathematics to them is just an application of "pure" mathematical truths to something else. And those who believe it is invented do not believe in "purity". Many modern philosophers consider the discovery/invention to be a false dilemma with neither metaphor adequately describing the genesis of mathematics.
    – Conifold
    Jun 20, 2020 at 7:18

1 Answer 1


Pure mathematics can and does stand alone as its own science.

It is the objective of pure mathematics to systematically express its content in a culture-free manner. If its truth content depends on the cultural context of its practitioners then it is assuredly not pure mathematics; in fact, if that is true then it does not qualify as mathematics at all.

Remember that the mathematical universe is intended to represent the logical consequences of sets of fundamental axioms. Seen in this way, mathematical discoveries most often involve the uncovering of connections between different pieces of that universe which were previously unknown.

Those "pieces" constitute inventions made by human minds, but they are almost always logically anchored to other parts of that universe and as such represent extensions added onto the edges of that universe.

Metaphorically speaking, once such connections between the inventions are discovered, the mathematical "scaffolding" gets quickly bolted together and the gaps between those sub-fields get filled in, and fundamental questions about the overarching structure of the mathematical universe can be answered.

This means that pure mathematics contains both inventions and discoveries.

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    "It is the objective of pure mathematics to systematically express its content in a culture-free manner." - what of the culture of mathematics itself? That is, would it not be closer to the truth that the objective of pure mathematics is to express it's content purely in terms of its own associated culture?
    – Steve
    Jun 20, 2020 at 13:47
  • Define for me that culture. Jun 20, 2020 at 17:30
  • Would I not just be defining mathematics and the philosophy of mathematics? That is, if we hold that mathematics is a cultural practice, then the definition of that culture is just the definition of mathematics. Cultural conflicts do arise - notably with computer scientists and programmers, who have been described as the engineers of mathematics, and often have significantly different opinions and approaches on various matters than mathematicians who are computer-illiterate.
    – Steve
    Jun 20, 2020 at 18:29
  • but there is no such thing as hindu mathematics, muslim mathematics, caucasian mathematics, chilean mathematics. And the basis for the field of mathematics is not culturally constructed Jun 20, 2020 at 18:34
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    That's because the contributions of all those cultures have been either integrated or discarded and now just go under one guise of "mathematics". You say it's not culturally constructed - I'm just saying mathematicians share a culture, and they construct the entire field in terms of that culture. The absence of any geographic differences doesn't prove there is no culture - only that such culture does not vary geographically (or not in any way you consider pertinent). And probably, anything that does vary you just declare "not mathematics", so we find "invariance" more like an axiom than fact.
    – Steve
    Jun 20, 2020 at 18:50

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