Except once axioms have been fixed, the resulting geometry will be the same wether you perceive reality via visible light or echolocation.
Maybe visualization is more helpful, but you can visualize things without seeing.
A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required.
From another point of view, math theories are congruent, and you can explain the same things without using geometry, but using algebra, or differential equations or number theory and so on. You certainly can explain any math concept using just modern set theory - maybe it is not the most convenient thing to do.
It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]
recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]
This is a quote from a book by Alexei Sossinsky, quoted here: https://onionesquereality.wordpress.com/2011/07/31/blind-geo...
How can you be sure?
I could see the layman understanding of geometry being quite different than ours, limited by their own visual system. But their mathematicians and our mathematicians would discover the same math, through different paths and perhaps with different strengths and weaknesses and a different path of educating budding mathematicians from the grade school level to the post doc level.
Can you get data on this from a Monte Carlo method of study on match play between sighted and blind players playing go and chess at distance and blind which requires a mental model of the board state and translations?
https://corecursive.com/035-bartosz-milewski-category-theory...
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?
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.
Why integers one might ask. For me, even rational numbers are such an approximation.
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.
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.
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.
http://www.math.pitt.edu/~bard/bardware/classes/0220/dkc.pdf
Ctrl-F for "2D" (2-dimensional creatures), although I can't find the "discovery", maybe I remember it wrong.
Btw, this was one of the mindfucks I learned studying physics. Other ones are:
- dimensional analysis: checking if formulas can be wrong, and more importantly, finding formulas (eg, period of a pendulum) just by analyzing units that can be involved (no constants obv)
- similarly, rate of growths in dimensions: why giants can't exist (bone resistance is proportional to section area and load (weight) is proportional to volume
- Conservation laws derive easily from symmetries (energy from invariance in time, momentum from invariance in direction etc)
Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.