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.
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)Location was not specified ...
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?
I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.
- Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3]
This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.
Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.
[1] https://www.quantamagazine.org/landmark-computer-science-pro...
[2] https://www.quantamagazine.org/does-time-really-flow-new-clu...
[3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...
There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will with weather forecasting.
Even if that weren't true you still can't get to Conway's Step 3, because "free will" could just be statistical noise, and not willed in any sense at all.
Is it though? I came to the conclusion that one could consider free will the self-realization of a person in space-time -- in other words, free will is the only and only choice one can make (at each decision they face), that they make because of who they are thus far.
In that sense, "free will" is the same as the "person" (or the person's essense) -- and would it make sense for it to be any other way? Randomness wouldn't be free will, and having a second entity (e.g. a soul in dualism) do the decision making just moves the question and degree upwards (from "how the person has free will?" to "how the person's soul has free will?")
If a person faced with a choice A or B could go either way, then I wouldn't call that really free will (even though for some reason that's what most people have in mind when talking about the subject).
Either a person is a concrete personality/being and that manifests through its choices (which means that when faced with the A or B dillema they can only chose one or the other based on who they are), or they have some degree of randomness in their decision making (which makes them less of a person in my eyes -- randomness is not tied to our person, it is, as the name reveals, random).
When some people say "free will" they actually mean "free (independent) from the person". But that's not a will then -- a will denotes something being tied to a person (a person is their will).
It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.
Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Therefore, while I will not support a metaphysical dualist basis for ‘free will’ it is clearly incorrect to dismiss an argument of such based on 1) an appeal to authority which is unsupportable (in that your claim about the position of a majority of ‘experts’ seems unsupported), and 2) that even if it was supported, that said appeal to authority is a valid refutation of an argument. Further, dismissal of an argument by choosing to substitute a contested definitional term (the meaning of free will) for one that is clearly not the same is not rhetorically valid.
Almost 60% Compatibilist, the remainder evenly split among three other options: https://philpapers.org/surveys/results.pl
> Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
You're assuming this is relevant. Turns out, it's not.
> Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it.
Nobody definitively understands free will. Some people conjecture it has certain properties, mainly incompatibilists. So far they have mostly been wrong.
> This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Experimental philosohy suggests pretty definitively that people's moral reasoning agrees with Compatibilism: https://www.researchgate.net/publication/274892120_Why_Compa...
> that even if it was supported, that said appeal to authority is a valid refutation of an argument.
The other poster provided no argument, they just made an unsupported claim.
A coin flip and a hand of cards are both random, but obviously cards are 'more random'.
This can be quantified - 'more random' is information complexity, or information entropy. (Two names for the same thing.)
If free will exists, then it must necessarily be a singularity of infinite information complexity, which is the same thing as a singularity of infinite randomness.
Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options.
Eg, the “natural numbers” are a recipe to make as many as you want (by taking the successor of the last one) but not an actual (as in existing) infinite collection — the only ones which exist are the ones you construct, and the “potential infinity” refers to the fact that you can always make a new one.
But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".
For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra.
However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it.
The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.
I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.
There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.
What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.
My point is something else: I don't know of a way to have a intuitionistic exterior logic and a classical internal logic.
> you can do intuitionist mathematics in classical mathematics
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough.
I am sure other HN users understand this stuff better.
The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory.
That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1]
[0] https://en.wikipedia.org/wiki/Intuitionism
[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...