I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode)
And really one of the biggest pieces of proof of this is that some of the earliest mathematics developed was planar geometry.
But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world.
This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.
But, this makes sense because we really don't ask the same question about human language. We almost never ask: is human language constructed or discovered?
>This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.
As I see it, mathematics is both discovered and invented.
We can model every existing thing in every possible world using math. Even if both the set of all things that might exist and all possible mathematical constructions are infinite, the later is larger. That's because we can also construct mathematical models of things that doesn't exist.
So it looks that from the set of all possible mathematical constructions, we extracted a subset that maps to objects in reality. That looks like a discovery process.
But we also constructed mathematical models of things that don't have corespondents in reality, so that much be more of an invention process.
Let's pretend for a bit that we forget all what we know and tomorrow we will start inventing things again. Or that a species of aliens start fresh on a planet.
The mathematic theories and notions we and the aliens might discover, build or invent might be different than the theories and notions we know today but would probably be equivalent. That suggests that mathematics just exists somewhere in its own world waiting to be discovered.
I too, subscribe to this view. In my mind, reality is nothing more than a vast integer variable field with interactions between some variables. All our mental models, from vision to math are attempts at approximating that complexity down to a level that our brain's limited processing capacity can handle...
Why integers one might ask. For me, even rational numbers are such an approximation.
This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge.
It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.
The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality is mediated by our brain's own idiosyncrasies and limitations of thinking.
It's no more surprising that external objective reality is mathematically predictable and describable than it is that our eyes can see shapes and some, but not all, light. That's the purpose of eyes and they tell us enough of what we need to know that we can survive.
Our brains are exactly the same thing- they tell us a story in a way which helps us to survive. Actual reality may be beyond our capacity to conceive of or worse, may seem like nonsense to us because it's aggressively illogical or specifically contradictory and reality therefore makes no "sense" to us.
>It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.
But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)?
I would say our brains do have limits. I can't well envision a 4D object. But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist. Even if I was a simulation that didn't even have a brain, much less eyes, the concepts I come up with would exist more than the flesh I incorrectly thought I had.
>>But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist.
I agree.
But this assumes logic and at least the principle of non-contradiction: not both A and not A - is how reality "really" is.
We can't get past the idea that it must be this way because only provable nonsense lies on the other side of this assumption. But our brains may be fundamentally unable to process ultimate reality and the nonsense a contradiction represents may be a statement not about reality but our brains, our thinking.
So logical contradictions aren't actually nonsense, they're the sound of us hitting the walls of what our minds can conceive of. All animal have such limits. We assume those limits look like darkness- stuff we can't peer into. What if they look like impossibility instead?
That's the point of view I'm entertaining here. I am not saying this is true. It certainly isn't useful or provable as far as I know, but it is possible.
The practical value of such an exercise, if it has any (and I think it does) is to twofold.
One seems to expand my imagination to the maximum extent pops me out of the assumptions that frame my thinking and this seeps into my thinking about things, technical problems, generally.
Two it confers humility and a certain openess and makes me less judgmental. So that, for example, when I hear or read people with claims to spiritual knowledge I don't automatically blow them off as crazy / bitter / ignorant because what they're saying "makes no sense".
Thinking and talking about Ultimate Reality capital U capital R, ought to fill us all with humility if we're being intellectually honest by our own standards. Yet, I find people totally lack that humility. They make huge pronouncements about Ultimate Reality which they can't really be sure of, and the effect this has on the world, and how we think of each other, and therefore how we treat each other, and even the effect on one's own mind, is one of diminishment generally.
Mea culpa, I was one of those people and I didn't like it.
If you get down to the core of knowledge and how we know something, this is what's really there, and it's good to be reminded of it.
https://corecursive.com/035-bartosz-milewski-category-theory...