All mathematical structure exist.

  • I
  • Thread starter vectorcube
  • Start date
  • #26
qsa
345
1
well, anti-platonism is a completely consistent view.


I was suggesting that, even he, believed in natural numbers are true and given. At least this is my understanding.
 
  • #27
I think it's pretty clear what's going on here. If we accept that mathematics transcends us and our universe utterly, the ultimate ensemble, and perhaps also those related ideas, are perfectly reasonable. If we don't, then they are not.

Some people here seem to accept that mathematics transcends us utterly, others don't. I like the idea that it does and I give it as a stipulation before entering into any philosophy where it is likely to be involved. That is not to say that I necessarily "believe" it, that would be dumb, it is clearly impossible to demonstrate something like this.
 
  • #29
apeiron
Gold Member
2,013
1
Some people here seem to accept that mathematics transcends us utterly, others don't. I like the idea that it does and I give it as a stipulation before entering into any philosophy where it is likely to be involved. That is not to say that I necessarily "believe" it, that would be dumb, it is clearly impossible to demonstrate something like this.
It is fair enough to take transcendance as an unproveable principle and then to consider what must follow. But why does it seem a plausible one in the first place, compared to the alternative?

To me, I prefer to assume that all is somehow one, all is connected, related. Once some essential aspect of reality is taken to be broken off and floated away, then a causal relationship becomes illogical, or at least paradoxical, mystical.

Why should I give up the notion that reality is a connected whole? The fact that reality is stratified seems obvious. So local instance is different from global principle.

But platonic maths, gods, mind, beauty, truth and goodness, psychic powers - every notion supported by claims of transcendance ends up just causing endless trouble.

Why should it still be anyone's preferred hypothesis?
 
  • #30
I don't see the supposition that mathematics transcends our existence as causing any trouble. Moreover, it does not require that a piece of reality be cut loose so as to float away. As I see it, what we call reality, along with all of the other realities of the ultimate ensemble are not cut off from each other, they are all subordinate to mathematics, tied together under that.

It is my preferred hypothesis because I see any alternative as belittling any conclusions that are ever made about anything. If I suppose that logic and mathematics are only as they are for me, now, then maybe someone else could come along and not only work with different starting assumptions and predefined rules, but come up with things such as "Your usually understood rules of addition and the definitions of the real numbers imply that 2+2=5."

The proposal that mathematics is not divorced from our experience and our universe is equally unprovable as the proposal that it is. Also, I think the fact that logic and mathematics as we understand them now work in every situation that has ever been envisaged is strong circumstantial evidence in favour of their universality.
 
Last edited:
  • #31
apeiron
Gold Member
2,013
1
I don't see the supposition that mathematics transcends our existence as causing any trouble. Moreover, it does not require that a piece of reality be cut loose so as to float away. As I see it, what we call reality, along with all of the other realities of the ultimate ensemble are not cut off from eachother, they are all subordinate to mathematics, tied together under that.
OK, so what is the nature of the connection between ensembles? This is the classic issue of platonism. How do the forms actually shape the chora in practice?

You say things are all "tied together" which implies some action on something's part. How does the transcendant actually achieve such feat?

I'm not saying there is absolutely nothing like the transcendant.

In arguing that "all is relationships", this is an ontology that also requires the "existence" of limits. Reality has its boundaries or event horizons. This seems to raise the question of what lies "beyond". The answer would have to be truly nothing. Or rather, only vagueness.

But this is not the same as a positive claim about things like gods or numbers standing in some abstract place beyond our concrete existence.
 
  • #32
They are not tied together in any material way, they can have no interaction with each other. Or more correctly, if ever two worlds have interaction with each other, they should be considered as one world, or mathematical object. The thing that connects them is mathematics and logic as we understand it now, I propose that it is valid everywhere (that applies to the everywhere of every contingency) and at all times.

As for how the ideas come to give form to things, I answer that it happened in the same way that our world was given form, whatever way that may be.

As for boundaries to reality, it depends on what you mean by "reality." The boundaries on our universe are those that are consistent with the initial conditions of the universe that can give rise to us as sentient beings exactly as we are. If you meant it in a broader sense, all possible universes, there are no boundaries, there are an infinite number of mathematical objects that can describe a universe so there are infinitely many universes. Similarly, some of those universes are themselves infinite.
 
  • #33
299
0
I don't see the supposition that mathematics transcends our existence as causing any trouble. Moreover, it does not require that a piece of reality be cut loose so as to float away. As I see it, what we call reality, along with all of the other realities of the ultimate ensemble are not cut off from each other, they are all subordinate to mathematics, tied together under that.
if modal realism is true, then the ultimate ensamble is a proper subset of modal possible worlds.
 
  • #34
apeiron
Gold Member
2,013
1
As for how the ideas come to give form to things, I answer that it happened in the same way that our world was given form, whatever way that may be.
But you can see that this is unsatisfactory. It is what makes me more interested in alternative approaches which do offer answers.

If you meant it in a broader sense, all possible universes, there are no boundaries, there are an infinite number of mathematical objects that can describe a universe so there are infinitely many universes. Similarly, some of those universes are themselves infinite.
That is a contention and so needs to be supported by some argument. And if you don't even have a speculative story on how maths/form entails physical universes, why should we place too much credence on the possible fruits of this (non)relationship?

Infinite math objects may = infinite actual universes. It is the idea de jour. But I am asking about the robustness of the eqivalence relation being claimed. Apart from the fact you can say it, why would we believe it?
 
  • #35
apeiron
Gold Member
2,013
1
if modal realism is true, then the ultimate ensamble is a proper subset of modal possible worlds.
Appreciate the fact you are now taking the more careful epistemic approach of qualifying "if X is true". That is really helpful to serious discussion (and I do find the possibility worth discussing simply because it is the extremal position of a particular line of thought).
 
  • #36
299
0
Appreciate the fact you are now taking the more careful epistemic approach of qualifying "if X is true". That is really helpful to serious discussion (and I do find the possibility worth discussing simply because it is the extremal position of a particular line of thought).
" is true" is metaphysics. It is not an epistemic notion.
 
  • #37
1,369
0
Example is fermat` s theorm. We now know that the theorm is true( because we have a proof), but conceivably, some alien from outer space would produce a proof that shows that theorm is true. The alien could never show that the theorm is falses even if they want to. Similarly, fermat` s theorm would be true even if fermat never formulated the conjuncture in the first place.
That confuses the theorem, with the evidence for the theorem. Evidence, facts about the world, are what is most often referred to as objective. The theorem describes something that may or may not be supported by evidence.

Proof is a mathematical concept, that generally has to do with logical consistency, not evidence.

Evidence is not 'proof' of anything. Evidence provides a basis for prediction.
 
  • #38
Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
12,121
159
Not sure I agree.

Re: the elephant. How can there be an objective statement about something only you can see? It is, by definition, subjective. Same with Santa, it is only agreed upon by the general populace that he wears a red suit. That's not objective.

How would you go about falsifying such claims?
These philosophical arguments are tougher than I thought they would be.
 
  • #39
apeiron
Gold Member
2,013
1
" is true" is metaphysics. It is not an epistemic notion.
And "if" - the actually relevant qualifying word here?
 
  • #40
299
0
And "if" - the actually relevant qualifying word here?
Does that matter?
 
  • #41
299
0
That confuses the theorem, with the evidence for the theorem. Evidence, facts about the world, are what is most often referred to as objective..

Did i say evidence?


The theorem describes something that may or may not be supported by evidence.
Empirical evidence? It is crazy to me why you would talk about evidence here. Math propositions are necessary true. As such, they cannot be falsified by evidence like any scientific theory.


roof is a mathematical concept, that generally has to do with logical consistency, not evidence.
Evidence is not the right word. What you want is deduction. Theorms are deduced from premises. What you say about logical consistency do little to explicated proof. I would say criterion are much more strict.
 
  • #42
1,369
0
Did i say evidence?
If it is objective, it exists independent of mind. Unless you base your mathematical axioms on some type of common existing 'evidence', then your axioms will be completely arbitrary, and so will your alien's axioms. In which case, you would come up with completely different theorems. Your theorems would untrue for your alien, and vice versa.
Empirical evidence? It is crazy to me why you would talk about evidence here. Math propositions are necessary true. As such, they cannot be falsified by evidence like any scientific theory.
Mathemathical axioms are definitions. Those definitions are based on human experience.
Evidence is not the right word. What you want is deduction. Theorms are deduced from premises. What you say about logical consistency do little to explicated proof. I would say criterion are much more strict.
Where do you get your premises and criterion?
 
  • #43
apeiron
Gold Member
2,013
1
Does that matter?
If something is true does tend to have a different meaning that something is true. So yes, you could say it matters.
 
  • #44
299
0
If it is objective, it exists independent of mind. Unless you base your mathematical axioms on some type of common existing 'evidence',
Again, there is no evidence. Math objects to not have any causal relation to physical matter.

then your axioms will be completely arbitrary, and so will your alien's axioms. In which case, you would come up with completely different theorems. Your theorems would untrue for your alien, and vice versa.
I get what you are saying, but the way you say it is wrong.

Platonism is the view that there are objective mathematical facts. Now, platonism is not without it` s problems. The most problematic( one would say the only problem) problem is the explication of how we come to know these mathematical facts. This is an epistemic problem. If mathematical facts exist, then they have no causal connection with the world, and thus, there is really no evidence.
Mathemathical axioms are definitions. Those definitions are based on human experience.
For platonist, math axioms are not definitions at all. The axioms are used to describe mathematical facts.


Where do you get your premises and criterion?
That is outside the issue. I rather we remain focus. At present, you seem to not know platonism, and i think you ought to read about it before you reply. I suggest you read about it, and ask me questions. That way, you can learn something.
 
  • #45
299
0
If something is true does tend to have a different meaning that something is true. So yes, you could say it matters.
What does that matter for the topic at hand?
 
  • #46
qsa
345
1
wiki quote

"It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematics program."

I think there is big confusion here between "compelling inevitability" and the "source". I think people get the idea because the source can debated, the "compelling inevitability" is not so compelling. even the "fictional" interpertation does not doubt 2+2=4

no matter what source, these "compelling inevitability" is what existence is made of.Is there any other really "compelling" entities we can count on.
 
  • #47
299
0
wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.
 
  • #48
qsa
345
1
wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.
I agree, but I was trying to give conclusion type statment to arguments. It would take too
much time to argue every little concept (philosophy does a good job in not closing issues and openning new ones). My interest is finding how reality works in the physics sense, but I use just enough pertinent philosophy(tammed in Einstien's word) to excute my goal. I learned that when I did my Master's Degree in UK; the stress is on research.

I wonder if you have any thought on the articles in fqxi site.
 
  • #49
1,369
0
Again, there is no evidence. Math objects to not have any causal relation to physical matter.
Useful mathematical statements like 1+1=2, are abstract representations of the physical observable world. If you are going to claim they exist independently, you need some sort of evidence to show that this is so. The axioms of modern mathematics are not random, they have a solid foundation in the physical, which is why they can describe the physical so well.
Platonism is the view that there are objective mathematical facts.
Plato was wrong. There is no higher reality of forms. Its not necessary, nor is there any evidence for such a thing. The ancient greeks were overly impressed with abstract thinking because their understanding of the physical world was so rudimentary. It was thought that the physical world was chaotic, ruled by the whim of the gods. They could use mathematics and geometry, which was logical and predictable, as a foundation.

But the reason mathematics was logical and predictable is because it is abstract and constructed. It was like the difference between living in a cave, and building a house. The latter was preferable to the greeks because they could design it to fit what they needed. Mathematics was designed and constructed to address certain needs, which is why it appears more solid than say the english language, which is more chaotic.
If mathematical facts exist, then they have no causal connection with the world, and thus, there is really no evidence.
Then they have no relation to this world, and are pure fantasy.
That way, you can learn something.
LOL. whatever.
 
  • #50
1,369
0
wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.
If you really want to learn something, don't rely on secondary sources.
 

Related Threads on All mathematical structure exist.

Replies
4
Views
984
Replies
1
Views
1K
Replies
5
Views
2K
Replies
1
Views
2K
Replies
6
Views
1K
Replies
21
Views
3K
Replies
2
Views
1K
Top