Just an interesting thing I found to think about. I think there are going to be many different interpretations of this question, but is there anything you would consider objectively true about our understanding of mathematics?
Agreed. Axioms are created, everything else is a property of those axioms and 'discovered.' Those basic axioms tend to be the most useful in our universe, but they were still 'created.' We often define or create other axioms, for example, taxicab geometries are imminently useful for modelling paths in cities. We created the rules for the taxicab geometry, as with Euclidian geometry, but we didn't create the properties that emerge from those rules. I think maybe the confusion comes from a misunderstood analogy: people think 'if you build a castle of LEGOs, you created that castle; aren't maths likewise created?' The misunderstanding is that mathematical theorems and properties aren't like the castle, they're like the potential to build a castle from those given LEGOs. The child created the castle, but she only discovered the possibility of building a castle, which always existed as a property of those LEGOs.
I think mathematics in terms of physics, because that's where I did my heavy learning. So when I think 'operator' I think in terms of QM operators, which are defined actions, instead of assumed properties. I see the two as separate, although I suppose they probably seem less so when viewed through a purely mathematical lens.
I don't know that I totally agree. Defining something is not really creating it as whatever it is already exists. I do somewhat agree in the sense that people create equations to explain things that happen in the real world. Nature doesn't give a fuck about equations. Also things like prediction of future problems and answers with equations people make with stats from present and past I feel has an aspect of creation.
Math is a lot like language. It describes the outside world in a way that other people can understand, provided you share the same language--or math. Math describes general truths about the world which can be applied to specific situations (e.g. if 2+2=4 then adding two apples to two oranges makes four fruits), but this is also true of some linguistic statements (e.g. if Dr Pepper bottles are maroon then the Dr Pepper bottle you're telling me about must be maroon). So I view math as sort of a special kind of language: one that consists of general statements which are universally true. Once it's reframed this way, it seems obvious that math was created. The relations which math describes were discovered, but we then created a system, mathematics, by which to codify those relations. N.B. Haven't actually read the article
I agree on maths being a human construct formed from the observations of the behaviour of the universe. There's nothing really "objective" since we can't compare this behaviour with respect to an absolute reference outside of the universe. But hey, it's managed to predict outcomes pretty accurately. I also prefer "creating" rather than "discovering" new concepts in maths. Going from the above points, maths is really more a tool for determining the properties of the universe rather than a natural phenomenon of it. It's a "creation" rather than a "discovery" to me.
OK! I think it's a difficult question because it's really quite meaningless. All of our logic, all of the way we think, is developed from our interaction with the universe, in very fundamental ways. I am in my bed. My bed is in my room. Therefore I am in my room. In this way our logic develops. Mathematics is the body of knowledge that grows from this logic. It is a human construct and it does such an amazingly good job of describing the universe because it grew from the structure of the universe. So when we find a surprising result and then find an example of that result in nature, we are even more surprised and get a feeling of being small surrounded by The Big Mystery. "Was the math there or did we make it?" ignores the fact that we ourselves are a part of the mechanisms of the universe. To me, creating and discovering are the same. I am just happy I can take pleasure in the process! (I suspect that the universe itself is an emergent phenomenum that comes from the simplest of rules, "1+1=2" or even simpler, like "1". It only really makes sense to me to think that the universe is just one particle.)
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Can human thought ever be considered to be independent of experience?Einstein remarked, "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"