Would Mathematics be Considered a discovery or an Invention?

[ISamson]

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?

 

[Antuntun]

I think that maths was discovery until Pythagoras discovered it but then people started inventing different ways of calculating maths and different theories about maths. So in total my answer would be that maths was both discovered and invented.

 

[ISamson]

I think that maths was discovery until Pythagoras discovered it but then people started inventing different ways of calculating maths and different theories about maths. So in total my answer would be that maths was both discovered and invented.

I agree with your opinion, but how can math be both a discovery and an invention?

 

[MarkFL]

I would say that mathematicians discover the truths encompassed by the systems of mathematics that humankind invented.

 

[ISamson]

I would say that mathematicians discover the truths encompassed by the systems of mathematics that humankind invented.

Wise words and language!

 

[fresh_42]

I'm a pure Platonist, "Platonism describes idea as prior to matter". I push the entire existence question into the heaven of ideas. Real world objects are discrete - nothing smooth is out there - at latest on the level of electron microscopes. And nobody has ever drawn a true circle. However, neither stop us from "knowing" what is meant and calculate with it. I even consider pieces of music as a real world manifestation of an idea of the perfect version of named piece. So all those things are discoveries to me, and the invention part of it is merely an insight, a short glimpse of the idea.

I recently counted ways to consider a derivative and found ten different descriptions of the same single formula. Those observations are a kind of evidence to my point of view, and not really suited to raise doubts. Nevertheless, in the end it is a matter of taste and a question about a preferred philosophical school.

 

[ISamson]

Yes, as I said in my original thread comment, I think that mathematics is an invention, because we invented it to accurately describe the universe. It is an invention that just happens to confirm observations so well, that it is the most important invention of the human kind, in my opinion.

 

[Nidum]

It's neither an invention nor a discovery . Just something that evolved over time .

 

[ISamson]

It's neither an invention nor a discovery . Just something that evolved over time .

Yes, however what did it evolve from? Can you explain yourself a bit more, please?

 

[sysprog]

Mathematical facts are discovered or as yet undiscovered, and mathematical procedures by which mathematical facts are established, are primarily invented.

However, if one is searching for a proof of something that he intuits to be or supposes or hypothesizes to be a mathematical fact, and he finds such a proof , he may regard the proof to have been by him discovered.

Some may consider him to have invented the proof, or to have discovered it, or to have invented or discovered the method by which he arrived at it, or may take the view that discovery and invention, while not the same thing, are not necessarily mutually exclusive categories

Some mathematical facts, e.g. those regarding complex numbers, are partly discoveries based partly on invention.

 

[Svein]

I have been for long wondering if mathematics would be considered a discovery about the universe or an invention of the human brain.

In my mind (and I am a mathematician) mathematics is a branch of philosophy. The difference from the standard philosophic systems is that mathematics is more open about its basic tenets (axioms) and rules of deduction.

Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess.
-- Robert A. Heinlein --

 

[fresh_42]

Mathematics can never prove anything. ...-- Robert A. Heinlein --

This is what I think about physics. The fact the apple fell from the tree doesn't prove it will do tomorrow, too. O.k. the likelihood is pretty high but it is still a likelihood. An example closer to reality might be the proton decay. Mathematics on the other hand actually proves statements in the clearest way possible to us: Given certain deduction rules and certain conditions, then a conclusion is true. If someone means by proof an absolute truth in what sense ever, then neither can, nor mathematics, nor physics, nor any other scientific branch. Absolute truth simply doesn't exist. To consider mathematics as a form of philosophy is actually what it had been through centuries. The view as a language in which physics and other sciences are written is a quite modern point of view which disguises the nature of mathematics in my opinion. Or to say it with another quotation:

What the difference between pure and applied mathematics is? There is none. One has nothing to do with the other.
-- David Hilbert --

 

[Telorast]

Rather than a discovery or invention, I would call mathematics a development coming from our interaction with the universe. Mathematics needs arithmetic, it needs theory and discrimination. My assumption is that mathematics originated from observing & relating arithmetic with physical situations. How many apples do I have? My 2 apples together with his 3 apples give enough that all 4 of us can have one each. The last apple can be divided...

Empirical geometric observations led to reliable calculations and theorems, which allowed cross-application from hand-tools to buildings to the stars. At every point along the way, the human mind carried math away from its tether to physical objects & situations allowing for both purity & application.

 

[ISamson]

Rather than a discovery or invention, I would call it a development coming from our interaction with the universe. Mathematics needs arithmetic, it needs theory and discrimination. I'm just speculating but I would think that mathematics came from observing & relating arithmetic with physical situations. How many apples do I have? My 2 apples together with his 3 apples give enough that all 4 of us can have one each. The last apple can be divided...

Empirical geometric observations led to reliable calculations and theorems, which allowed cross-application from hand-tools to buildings to the stars. At every point along the way, the human mind carried math away from its tether to physical objects & situations allowing for both purity & application.

Yes, but I would rather say that mathematics is a development made by humankind to accurately describe our surroundings, as I have already mentioned. We have adapted mathematics and fit it with physics to dig deeper and deeper with our understanding of the universe.

 

[Telorast]

Yes, but I would rather say that mathematics is a development made by humankind to accurately describe our surroundings, as I have already mentioned. We have adapted mathematics and fit it with physics to dig deeper and deeper with our understanding of the universe.

I agree-- just saying it allows for more... deep understanding isn't always unambiguous. Also there can be misinterpretation, and there's my personal favorite, castles in the air, which are beautiful without having any particular function. (I've been called a space cadet on more than one occasion.)

 

[jim mcnamara]

What you are discussing is whether or not a Platonic view applies to mathematics. Platonism says:

the theory that numbers or other abstract objects are objective, timeless entities, independent of the physical world and of the symbols used to represent them

In other words we discover, not invent, mathematics - if you take the Platonic view. Most of the mathematicians I worked with at a National lab from my view as an outsider were:

1. Platonists
2. "Closet" Platonists - their language use betrayed their beliefs, but they never, AFAIK, said that objects existed independent of "discovery" or elucidation.

I asked quite a few of them over the years.

This is an old discussion, BTW. And it is, IMO, completely philosophical. Not really a good fit for PF. @fresh_42 is one of the keepers of the keys for this area.

 

[SSequence]

...
This is an old discussion, BTW. And it is, IMO, completely philosophical. Not really a good fit for PF. @fresh_42 is one of the keepers of the keys for this area.

I do believe that it is the discretion of mods to close of course. As I understand though, this kind of debate might be more closely related (or interesting possibly) to a set theorist or logician(from area other than set theory) than perhaps a working mathematician from other areas.

Even a logician such as late Solomon Fefferman has written at one point along the lines of "what is a real number anyway?" towards the end of an article, while also raising important concerns regarding more philosophically troubling issues such as CH. So I think logicians do tend to take these kind of topics more seriously (sometimes ofc, not always).

I was going to add a response based upon some basic issues that I see (quite briefly and without taking any opinionated side). But indeed there have been few threads similar to this before (in the last year or so). So, in any case, if mods want to close this, I guess it is up to their discretion.

So I guess I will first wait to see what fresh_42's opinion is about this thread (regarding a possible closure).

 

[Telorast]

... I do believe that it is the discretion of mods to close of course. ...

Or it could get relocated to General Discussions section

 

[SSequence]

Or it could get relocated to General Discussions section

I do not think this belongs to "general discussion". Because a lot of the associated issues are closely related to strictly mathematical topics (where there is also subtelty involved). I believe one can get some sense of the issues (just as a general roadmap, after that details start to become more important) involved even without going to the level of detail all the way, since it gives some understanding of wider perspective.

As an example, I wouldn't want to study constructivist stance in more detail before I have a sense of potential shortcomings and strengths associated with it (and same holds for many other topics of study). Ofc that's just my personal view and many might reasonably disagree. Admittedly sometimes I have becomes overly stubborn with selection of topics to study, which can be personally detrimental at times in terms of learning (but that's another thing).

Anyway, let's wait for mods to decide.

 

[TeethWhitener]

I always just assumed the axioms were invented and the theorems were discovered.

 

[fresh_42]

@jim mcnamara is completely right. Yes, it is a really, really old discussion. Yes, most mathematicians are Platonists. Yes, they almost never fit into its narrowest definition so it might well be called closet-Platonism. And finally, yes it is about philosophy. But I have some headaches calling it as such, as I seriously doubt, that we will be able to debate on a scientific philosophical level. So far the discussion was very civilized and the posts have been more of a gathering of personal opinions, than it was a philosophical discourse. I think there is a need, to discuss this from time to time as most mathematicians are sooner or later confronted with the subject. I remember we had a thread not that long ago, which dealt with the question, whether $0$ is real. That lasted if I remember correctly longer than fifty posts or so. Compared to this, the current thread is refreshingly calm and reasonable. And by the way, there are similar questions in QM which are frequently discussed, namely about the interpretations of modern QFT. So as long this thread doesn't get heated by ideologists, I thought it would do no harm. The philosophy rule is mainly to avoid endless debates on nothing, resp. personal taste, and argues about right and wrong. In this sense the thread until now is far from ending up there. The option to close it, however, is always given, and maybe I'll be overruled by other mentors.

 

[SSequence]

A lot of the discussion regarding this topic sometimes revolves around "geometry". Since I don't have a reasonably informed opinion about that, I will skip that.

To me one of the biggest issues is "What kind subsets of ℕ are to be admitted as acceptable?". Quite honestly, this is one of the reasons why I have become more interested in mathematics in the first place (in last year or two). One would note that the question is equivalent to "What kind of reals are to admitted as acceptable?".

Basically a lot of opinions and varieties regarding math indirectly are related to this question. My current perspective (and obviously quite incomplete) perspective is that this takes one to the following three points:
(1) Strict Constructivism / Intuitionism
(1a) Strict Constructivism: As I see it, strict constrctivism is markedly different from classical math (to the point that it can't be made compatible). This is not to say that one can derive very many interesting theorems etc., but the understanding of basic idea of "reals" is radically different in constructivism.

Of course a classical mathematician might (and probably will) raise the objection that the set of objects admitted as "reals" are just too narrow (and hence don't represent the "actual continuum"). As I see it, if one doesn't pretend that one is dealing with some "actual continuum" (whatever that is supposed to mean) in constructivism, it is very close "discovery".

There is still some space for variations though. For example, as I re-call, Markov assumed that every program either halts or doesn't halt (essentially LEM over halt set). But someone could be more conservative than this (and not assume that).

(1b) Intuitionism: I don't mean intuitionism in a some strict sense of variety here (because there would be many varieties), but in a very broad sense. For example, Brouwer made the distinction between "lawless" and "lawlike" seq., which I don't agree with (because my feeling is that such kind of distinctions can be made for "relative analysis" but not for "absolute analysis").

Nevertheless, the phrase "operations of human mind" most closely reflects my own views. And ofc this is why my personal hope/wish is that if we interpret intuitionism in a very broad sense, there might be hope for its reconciliation with classical math (to a fairly good degree). But indeed that's not how mathematics operates (not on hopes but what actually is the case). It is possible that such a reconciliation might not be possible in principle though.

(2) Objects below Ω/ψ
Now one might also bypass the previous discussions entirely (as I understand most mathematicians do) and focus on what makes "sense" (at least as long as one is always willing to assume a yes/no membership for reasonably well-defined subsets of ℕ).

Now it seems to me the bigger the set of ordinals below Ω one assumes one can go on to define more and more strong hierarchies of "subsets of ℕ". For example, in infinite program models the longer a program runs the bigger the collection of sets it can produce.

"Ultimately" I believe one would, at some point, have to face difficult questions regarding "bottom-up" constructions, "reasonably well-defined" constructions (and hence also the issue of discovery/invention to an extent). And such questions I think might depend on what kind of view one has regarding sets.
It seems to me that some of the "multiverse" (as opposed to "platonism") view regarding maths comes from here. A fairly relevant link (quite advanced, probably good as a reference for long term).

To be honest though, I have quite a lot of difficulty reconciling this with the possiblity of ℝ not having cardinality aleph_1 and how that's "really" possible (and maybe learning set theory properly might help alleviate this kind of difficulty to an extent ... though I really doubt it would change my own views completely ... perhaps partially though). There seems to be some aspect here where I just can't seem to relate to classical thinking.

(3) Objects above Ψ/Ω
Once again the issue what might count as bottom-up in a reasonably way or not would come up it seems. And this might interrelate to (2) in a complex way it seems.

==========

And lastly one has to face the difficult questions such as:
(i) How much and exactly what assumptions we have to make about the "reals" to derive such and such theorem (in analysis for example). Of course I don't know anything about this.

(ii) If truth/falsity of some number-theoretic statements is derived using some subtle assumptions regarding (2) and (3) above, what are to we make of it? Surely we could dispel the subtle assumptions and assume nothing about the truth/falsity of the corresponding number-theoretic questions (but the difficult question exactly is when we have to take a stance).P.S. To summarise the whole discussion in few sentences, it seems that one can keep a lot of the description of mathematical ideas very well-grounded if one is very clear about the assumption regarding what collection of sub-sets of ℕ (and hence exactly which reals) one is talking about (shouldn't matter if the collection is very large). But it seems that the generality of "real continuum" will be lost then.

 

[SSequence]

Also some distinctions that are not often made clear often enough (and sometimes contributes to unnecessary confusion it seems). We have the following three aspects:
-- Physical World
-- Mind
-- Mathematical World

It isn't quite clear to me what the "multiverse" perspective is. It seems like that it is based on the idea perhaps that there are "different" mathematical worlds and we don't know which of those happens to represent the current "physical world" (assumption being that some mathematical world does correspond to the physical world?). Nevertheless, I am not quite clear about it.

The "formalist" perspective is that the mathematical world is without meaning in the same sense that the physical world is (as in postmodernist philosophy etc.).

Also it seems to me that the different between "platonist" and someone who is not a "pure platonist" is that the former separates the mind completely (or to a large degree) from the mathematical world, while the latter insists a strong relation between the mind and the mathematical world (to what extent obviously would depend on one's view).
There is no difference of view on objectivity of math. And also the "physical world" seems to be secondary in both cases.

But of course all that should just be taken lightly as guide rather than some absolute way of categorisation.

 

[Aufbauwerk 2045]

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.

 

[SSequence]

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.

That's very close to what I think. My main point of post#22 (which I am afraid might have been lost due to length) was simply that if one is willing to set aside constructivist/intuitionist objections, much of maths can atleast in principle be put or modified a bit to fit in a very clear form. So that in such form it would be fairly clear that it is a discovery.

However, that's if one doesn't include uncountables or continuum("real continuum" I mean). Because there are lot of subtle issues present with these concepts, where the divisions can be less clear (but conversely, they also seem to be very powerful concepts that give one a uniform language to communicate).

 

[fresh_42]

Here's another quotation of an opinion I don't share, but I'd like to cite for the sake of completeness:

The integers have been created by God, everything else is human work.
-- Leopold Kronecker --

 

[lavinia]

Here's another quotation of an opinion I don't share, but I'd like to cite for the sake of completeness:

The integers have been created by God, everything else is human work.
-- Leopold Kronecker --

According to the Wikipedia article David Hilbert argued that "mathematical truth was independent of the existence of God or other a priori assumptions.[25][26]"

 

[fresh_42]

According to the Wikipedia article David Hilbert argued that "mathematical truth was independent of the existence of God or other a priori assumptions.[25][26]"

I don't think we have the right to judge on Kronecker's religiosity, and even more, it cannot be seen without considering the time it has been said in. I'm sure he used "God" simply as an indication of something "natural" in contrast to something "artificial". Plus, there is no proper translation for the original quotation: "Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk." (L.Kronecker) "Der liebe Gott" is a language people use when they talk to children and want to indicate some superior reason, it is far less strict than the English "God". So it is obvious (in the original version) that he meant to distinguish natural and artificial, rather than making a statement about God. Furthermore, the modern sensitivity after the secularization of science wasn't present when Kronecker lived. I've never seen someone analyzing Einstein's "Der liebe Gott würfelt nicht." under the aspect of religion. So I think we must not do it with Kronecker.

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.

 

[lavinia]

Also some distinctions that are not often made clear often enough (and sometimes contributes to unnecessary confusion it seems). We have the following three aspects:
-- Physical World
-- Mind
-- Mathematical World

It isn't quite clear to me what the "multiverse" perspective is. It seems like that it is based on the idea perhaps that there are "different" mathematical worlds and we don't know which of those happens to represent the current "physical world" (assumption being that some mathematical world does correspond to the physical world?). Nevertheless, I am not quite clear about it.

The "formalist" perspective is that the mathematical world is without meaning in the same sense that the physical world is (as in postmodernist philosophy etc.).

Also it seems to me that the different between "platonist" and someone who is not a "pure platonist" is that the former separates the mind completely (or to a large degree) from the mathematical world, while the latter insists a strong relation between the mind and the mathematical world (to what extent obviously would depend on one's view).
There is no difference of view on objectivity of math. And also the "physical world" seems to be secondary in both cases.

But of course all that should just be taken lightly as guide rather than some absolute way of categorisation.

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?

 

[lavinia]

I don't think we have the right to judge on Kronecker's religiosity, and even more, it cannot be seen without considering the time it has been said in. I'm sure he used "God" simply as an indication of something "natural" in contrast to something "artificial". Plus, there is no proper translation for the original quotation: "Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk." (L.Kronecker) "Der liebe Gott" is a language people use when they talk to children and want to indicate some superior reason, it is far less strict than the English "God". So it is obvious (in the original version) that he meant to distinguish natural and artificial, rather than making a statement about God. Furthermore, the modern sensitivity after the secularization of science wasn't present when Kronecker lived. I've never seen someone analyzing Einstein's "Der liebe Gott würfelt nicht." under the aspect of religion. So I think we must not do it with Kronecker.

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 piece 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.

Interesting perspective.

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

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. Another way to put it would be that God is not contingent but everything else is contingent on God. I get the impression from spending a little time in an ashram that Hindus think of God in this way. Christian philosophers as far back in time as the Middle thought of God in this way. One might restate this as 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. Mathematics is not predicated on anything. This seems pretty radical.-The argument that mathematics is merely a tool or language to describe the empirical world seems to imply that mathematics is contingent on empirical observation. I don't think Hilbert would have bought that.

- If there is a unifying substrate to reality, then all things are natural. To distinguish between natural and human work is to say that human work is not natural which suggests that humans are not part of the universe. Is this what Dedekind meant?

 

[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.

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

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.

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

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.