High School Would Mathematics be Considered a discovery or an Invention?

[lavinia]

Sorry for the wrong prefix, but it is the most appropriate one that I could have chosen.
I have been for long wondering if mathematics would be considered a discovery about the universe or an invention of the human brain. If it is a discovery, then math is the universal language of the universe. If it is a discovery of the human brain, then we just invented it to accurately describe the universe and our surroundings.
What do you think?

Mathematics like natural phenomena are discovered.

 

[fresh_42]

Interesting perspective.

Can you explain though what the difference between God-given and Natural is? I don't see an obvious difference.

I don't see one either. What is now attributed by "natural" and "physical world" or even "meta-physical world" was formerly noted as "God given". It is a change of habit through the centuries and not necessarily a change of meaning. At least I see it this way. As soon as someone speaks about God nowadays we are opposing it with some sort of Pavlov reflex. It has been far more common in earlier times. I've often found the answer "it is as it is, that's nature, resp. universe" if people asked why questions. An old fashioned wording would have been: "because God made it this way".

I am not referring to the idea of God as an alpha superhero - but rather as a substrate for existence upon which all things depend. I get the impression from spending a little time in an ashram that Hindus also think of God in this way. Another way to put it would be that God is not contingent but everything else is contingent on God. Christian philosophers as far back in time as the Middle thought God in this way. One might say that God is that of which all things are predicated but which itself is not predicated of anything else.

From this point of view, Hilbert seems to be saying that mathematics is not contingent - not even on God. It is not predicated on anything. This seems pretty radical.

Not sure I know Hilbert's personal views well enough to comment on them. His (failed) program was in my opinion an attempt to achieve something fundamental, independent on all human influences, was it perception, interpretation, invention, language or simply philosophical existence or even a man made concept like God. The Wikipedia entry on Hilbert has some nice contributions to this discussion, especially as both men - Hilbert and Kronecker - have had a quite extrovert character along with some strict opinions.

Of interest and far more Earth bound will be the solution to $P = NP$. To me this is the real question: Are there inherently difficult problems, or haven't we simply found an answer to them. But this is a different discussion.

 

[Janosh89]

Of course , as infants we all (or nearly all) learn to count on our fingers . The pentadactyl terminations at the end of our
forearms mean that most humans are predisposed to the decimal system.
Early (valve) computers used ternary - e.g. Collosus , now superceded by binary.
One of those coincidences that DNA chromosomes have a 10-fold symmetry when viewed longitudinally.
Just a thought really, maybe we should stick to the idea that the axioms mathematics are built on ARE
strictly impartial.
I know ,of course, that mathematical induction & reasoning is largely number base "transparent". OK, 100%
transparent!

 

[SSequence]

Mathematicians generally view mathematics as unified rather than as separated into different mathematical worlds. I personally have difficulty even understanding what it would mean for there to be more than one mathematical world. Could you elaborate on that?

Well my formal background is computer science (though I never liked it ... obviously logic and metamath is seemingly more interesting for me), and my knowledge of logic and set theory is very little (so I won't invoke any of them directly). So I would give an answer based upon few things that I have an idea about.

Consider the set of ω-tape ITTMs. Given a blank tape an ITTM(that halts eventually) can go very far (and some will go further than the others). However, the maximum that the set of ITTMs can go has to be a countable. We can simply show this by giving an enumeration of the ω-tape ITTMs [EDIT : Actually the countability argument will have to be more sophisticated than this. If we define a gap as a position where no program can halt then one would also have to prove that the gap widths are always equal to a countable ordinal].
Before these programs, of course some of the ordinals involved were also defined before (purely in terms of logic). Using countability arguments (or similar) one could conclude that the involved ordinals were countable.
To some extent, before these kind of arguments is what I meant partially by the "uniformity of arguments" provided by the ideas such as uncountable.

But such programs also of course also generate a rich collection of subsets of ℕ. The interesting thing is that the collections formed are quite robust (it would be hard/impossible to think of a subset of ℕ outside the collection without explicitly thinking of a more powerful model or direct diagonalisation kind of thing).

Since you can't give an explicit well-order for Ω in terms of ℕ (by definition), it seems to me that a lot of views of largeness of world below Ω would inevitably come down to atleast some subjective opinion. And with this would also come the problem of collection of reals that we "really" admit (and how do we decide it). I could be wrong, but it seems to me that a lot of subjectivity part comes in here.
And this is without considering the case of ℝ not having cardinality aleph_1. How would we adjust this(card(ℝ)≠aleph_1) with our intuitive understanding that the collections of reals formed in bottom-up manner (going to arbitrarily large countable α's) are incredibly robust? I will admit my ignorance here tbh (maybe learning more set theory would help).

Regarding your point regarding mathematicians holding the "multiverse" view (I also have difficult time imagining it though), you might also search for it and I think you would find a lot of material of varying sophistication/level regarding it.

I would say that one can reasonably justify a lot of math (that would look a lot like traditional continuous math) to a very high confidence (essentially discovery) by restricting oneself to certain levels (but it seems that would also mean potentially missing upon some important ideas).
That's ofc ignoring the concerns in (1a) and (1b) of post#22. And even with enormous restrictions of (1a) it seems a fair amount continuous-like math can be done (analysis for example) ... just a little differently. One can find interesting literature on it.

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One idea that one "might" try (I don't know whether it is of trivial nature or not) is that try defining "reasonable-ness" of a function based upon the idea that, on top of a mathematical definition, there are at least some possibilities that one should be able to rule out reasonably. As an example consider the predicate Equivalent:ℕ²→{0,1}. Equivalent(x,y) is evaluated true if program corresponding to index x and index y both compute the same function (and false otherwise). Even though algorithmically one can't evaluate this function for all inputs, we do know that we can easily rule out some "possibilities".

That is at least for "some" inputs we can do better than "the answer is either 0 or 1". For example just choose a few programs that are trivially non-equivalent or equivalent and write down 0 or 1 for those specific inputs. In some sense it seems that some of our confidence regarding a "good definition" of a subset of ℕ comes from being able to rule out atleast some possibilities.

But really, I am just putting this as a thought. I don't have a good idea whether the idea can made to turn out to be trivial(not useful) or interesting.

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As a historical aside (since a lot is being mentioned in this thread), you might want to see this:
Comments on the Foundations of Set Theory --- Paul Cohen

Edit:
I realized upon a bit of searching that lot of Physics related links come up. So I am posting a link (one is much less likely to find any elementary or intermediate level discussion about it), mostly so that some of the key terms are easier to identify:
http://www.logic.univie.ac.at/~antos/wp-content/uploads/2015/01/MCST_Synthese_Complete.pdf

 

[ISamson]

My basic or naive view is that mathematics is part discovery and part invention. For example, I would say that the fact that pi is irrational is a discovery. I think that any alien civilization that came up with the definition of a circle as the set of points equidistant from a given point would discover that pi is irrational.

I would say using radians to measure angles is a discovery. But using 360 degrees for the circle is an invention, which goes back to ancient astronomers who realized the year is approximately 360 days. On another planet the circle might consist of 400 degrees, based on the length of their year. But a radian is a radian everywhere in the universe.

Of course one might say the mathematical circle is itself an invention, in the sense that it's a abstraction from reality produced by our brains. In that sense we could say that all of our knowledge is an invention of our brains.

For me this leads back to our brain, which is the real mystery, since it seems harder for us to understand our own brains than to figure out calculus, number theory, quantum physics or relativity. This is very discouraging to me. We can understand enough to come up with amazing technologies, we can understand a lot about the atom, but we lag way behind in understanding our brains. I think this is the situation that led me to AI via physics.

I agree. Mathematics is sometimes a discovery and sometimes an invention. But I am mostly with invention, because (as I have said in my earlier posts) we have invented mathematics to describe our surroundings accurately and efficiently. So, any other alien civilisation might not use anything similar to our mathematics to describe what they see.

 

[fresh_42]

So, any other alien civilisation might not use anything similar to our mathematics to describe what they see.

Wouldn't they necessarily have to arrive at the nature of primes, $\pi$ and $e$ as well? And maybe also at some similar logic systems and eventually similar results on decidability? I cannot see that any of our basic concepts $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ are "unnatural". The step to group theory, analysis or algebra seems to be equally natural to me. And one of the properties of mathematics is its growth and adaption. In the end, all mathematics is basically the same as to start with $\mathbb{N}$ and find $\mathbb{C}$ by continuous generalizations and adoption of new findings or needs. I think as universal as a hydrogen atom is, as universal are the elements of mathematics.

 

[ISamson]

Wouldn't they necessarily have to arrive at the nature of primes, $\pi$ and $e$ as well? And maybe also at some similar logic systems and eventually similar results on decidability? I cannot see that any of our basic concepts $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ are "unnatural". The step to group theory, analysis or algebra seems to be equally natural to me. And one of the properties of mathematics is its growth and adaption. In the end, all mathematics is basically the same as to start with $\mathbb{N}$ and find $\mathbb{C}$ by continuous generalizations and adoption of new findings or needs. I think as universal as a hydrogen atom is, as universal are the elements of mathematics.

I totally agree, but they might not necessarily use numbers for example.

 

[Janosh89]

Atomic Number ,or whatever they(?) would call it, would still be an observable method of counting of sorts.
1+1=2 ; 1 proton + 1 proton → 2 protons [ H⁺ + H⁺ ⇒He⁺⁺ where ⇒ indicates nuclear fusion].
I suppose that one could envisage a universe where irrational numbers represented the nuclear content of atoms..

 

[jbriggs444]

Atomic Number ,or whatever they(?) would call it, would still be an observable method of counting of sorts.
1+1=2 ; 1 proton + 1 proton → 2 protons [ H⁺ + H⁺ ⇒He⁺⁺ where ⇒ indicates nuclear fusion].
I suppose that one could envisage a universe where irrational numbers represented the nuclear content of atoms..

Put it in operational terms. If one proton corresponded to $\sqrt{2}$ baryon elements and one neutron corresponded to $\sqrt{2}$ baryon elements, what observable features of the universe would have to change? Answer: none at all.

 

[Janosh89]

I don't intend to put it in operational or explicit terms. It was meant as a continuation of previous posts regarding
NUMBER.. Thanks for your response,anyway, . I "know nothing" [reference to your avatar] and I certainly would
not post it here in Gen. Maths even if I did...?

 

[jbriggs444]

I don't intend to put it in operational or explicit terms. It was meant as a continuation of previous posts regarding
NUMBER.. Thanks for your response,anyway, . I "know nothing" [reference to your avatar] and I certainly would
not post it here in Gen. Maths even if I did...?

The point I was trying to make is that whether the value you choose to label as 1 in your system of natural numbers happens to be $\frac{\sqrt{2}}{2}$ baryons, $\pi$ apple pies, 12 flying purple spaghetti monsters or one set containing an empty set does not change the nature of the resulting system of natural numbers. The choice has no physical consequences. It does not change the natural numbers. Nor does it change quantum mechanics.

 

[SSequence]

Hilbert was probably far closer to Platonism as seemingly Kronecker was. In my opinion his program alone can be seen as an attempt to find mathematical truth in Plato's heaven of ideas. And I'm not sure whether we have made our peace with it's failure. To me it often seems, that we just have learned to live with it and the only difference to physicists and the quantum world is, that mathematicians don't speak about it and pretend everything is fine. Zorn and Gödel have become something for logicians and the rest don't bother.

I wanted to comment on it a little bit, but I was unsure how to do it in a more coherent manner (without diverging too much). First a little off-topic comment about Kronecker's quote. I remember reading a more extensive version of it where he basically says something along the following lines. What he was saying that if he had enough time (which he didn't) from his main mathematics, he would independently develop an entirely different set of ideas from scratch.
I tried to find that extended version but somehow I just can't find it now. I have learned that the original quote is also supposed to be recorded/taken from some lecture. So I am not sure whether the extended version which I read was correct or not.

Secondly, as I understand, there are basically the following ways to see the problem (as we go down, roughly speaking, the philosophical importance decreases and practical importance increases):
(1) Saturation Problem (Generalised Church Thesis)
(2) Church Thesis (Bounded Memory)
(3) Efficiency of Algorithmic CalculationsThe question (2) is settled in an absolute decisive sense (and hence also the absolute nature of incompleteness in the "bounded memory" setting). This is as long as one doesn't conflate (2) (a mathematical statement) with Physical Church thesis (a physical statement). There have been few proofs of (2) based upon axiomatisations (I don't know much in the way of details). Though perhaps not everyone might agree that a given axiomatisation is absolutely convincing.
But actually, I still believe that the truth of (2) is absolutely decisive (at any rate). I say this because of the growth functions that are formed as a result of the underlying hierarchies (by extending unsolvability). And also based upon how our natural understanding of "limits" of (transfinite) iteration (ordinals that is) happens to coincide with its recursive counterpart (that is using the symbols of ℕ only). So the words "bounded" and "recursive" happens to coincide exactly in this context. This is a point I happened to stumble upon a few years ago by "trial and error" before I first saw the term "ordinal" being used. But I don't know what would be the best way to make this particular point more formal.

Though I am not trying to downplay the importance of precise axiomatisations at all (in case it seemed like this), the two points in previous paragraph leave no doubt whatsoever in my mind that (2) is correct.

==============================

https://www3.nd.edu/~cfranks/frankstennenbaum.pdf
Specifically I wanted to quote the following part (on page-9):
"We might be inclined to doubt the finitist character of the ‘transfinite’ induction [through ε₀ used in his proof of the consistency of Peano Arithmetic (PA)], even if only because of its suspect name. In its defense it should here merely be pointed out that most somehow constructively oriented authors place special emphasis on building up constructively . . . an initial segment of the transfinite number sequence . . . . And in the consistency proof, and in possible future extensions of it [to theories stronger than PA], we are dealing only with an initial part, a “segment” of the second number class . . . . I fail to see . . . at what “point” that which is constructively indisputable is supposed to end, and where a further extension of transfinite induction is therefore thought to become disputable. I think, rather, that the reliability of the transfinite numbers required for the consistency proof compares with that of the first initial segments, say up to ω², in the same way as the reliability of a numerical calculation extending over a hundred pages with
that of a calculation of a few fines: it is merely a considerably vaster undertaking to convince oneself of this certainty . . . ."

Not counting some more precise technical details involved (which I obviously don't understand), one can still get a fair sense of the paragraph.