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    We discovered bananas but invented the word 'banana'. Isn't mathematics the language that describes the nature of things? That nature exists 'outside our heads' but surely the language is invented?

    (Original post by ChaoticButterfly)
    This seems sensible to me. But it leads to the conclusion that the concepts exist outside our heads and are "real" which I find confusing.

    How can you empirically find the existence of math concepts? Sure we can find how maths concepts fit in describing nature, see physics etc, fractals describing how plants grow is an example. But does that prove that there is an underlying maths that controls the plants growth on a deeper level? Is mathematics the fabric made by a deistic god who set the universe in motion? Is the universe made of maths?
    Now you are thinking! The problem with today's highly specialised teaching and youth is that they tend to fall into a type of logical positivism/scientism which says that all knowledge is either analytically true (true by definition) or empirically verified. This has been shown to be self defeating since the 1950's.

    However maths (and logic) were the biggest issues for scientism. It's widely accepted that we don't know or confirm maths empirically. Neither can science be used to justify maths - it's the other way round! Science has to assume the truth of mathematics in order to work. The same goes for logic (anyone who has read Bertrand Russell will know how close logic and maths are).

    It seems manifestly bizarre to argue we invented maths. We couldn’t change the outcome of the vast majority of maths if we wanted to. More over, it absolutely seems to be mind independent. Pythagoras theorem was true before humans were alive, and so it will be after they all die. Again, we couldn't alter geometrical proofs even if we wanted to and yet the world is written in maths. Geometry is the foundation for any possible mechanical engineering for example.

    Posted from TSR Mobile

    To know whether Mathematics is invented or discovered, we would have to know whether numbers are invented or discovered, because mathematical theorems result from the properties of numbers. So, for example, Pythagoras theorem is easily understandable when thought of in purely geometrical terms, but what it does is show that the square of two numbers is equal to the square of some other number (whether whose square root is integer or not). It is only when expressed geometrically that one can appreciate the symmetrical nature of the theorem.

    Comparing its steady advance to that of natural sciences, or even philosophy, one might easily characterise mathematics as an a prior inquiry, which demands no empirical enquiry, no observation of the world in order to succeed in establishing truths. But what does mathematics have to say about reality when it comes to describing the constituents of reality? Can it describe what matter is made of or what an atom looks like?

    Even though these questions seem remote from the subject-matter of mathematics, they say a lot about what numbers are, despite various attempts of mathematicians to place mathematics (or numbers) on purely logical foundations.

    Both. You invent the definitions, axioms and systems, but then you discover what their logical consequences are.
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