From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof.If mathematicians say something has mathematical existence, then it does. There is surely no objective standard. It's a mistake to believe that there is. — fishfry
Infinity is a principle that arises while reasoning from first principles, such as in mathematics, but not while experimental testing, such as in science. Furthermore, the models for number theory and set theory are never the physical universe. These models are collections of formal language strings. They are 100% abstract only.
Last but not least, you would not be able to prove anything about infinity in the physical universe, because you cannot prove anything at all about the physical universe. We do not have a copy of the Theory of Everything of which the physical universe is a model. Hence, there is no syntactic entailment ("proof") from theoretical axioms possible about the physical universe. — alcontali
A thing has mathematical existence when a preponderance of working professional mathematicians say it does.
— fishfry
So are you saying that when Georg Cantor first defined infinite sets ca. 1871 and there was great resistance among the world's mathematicians, infinity didn't exist yet? — Daz
Certainly mathematics is a social enterprise. And what constitutes a proof is a kind of consensus among those who practice mathematics. However, when I discovered last night a fact about attracting fixed points in polynomials that minor discovery immediately assumed mathematical existence, regardless of whether it is publicized. And it is possible someone else had arrived at this trivial conclusion, so it might have had mathematical existence already. But, in the larger social scheme there is a kind of mathematical existence based upon an agreement that a revelation is important. — jgill
From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. — aletheist
Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. — aletheist
Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof. — aletheist
For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. — aletheist
Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously.I think that's limiting. It puts trivial conclusions derived from meaningless axioms on the same level as the deepest results. — fishfry
It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers."No number theorist believes that Fermat's last theorem is merely a theorem that falls out of the axioms of set theory. Wiles proved that FLT is true. True in a way that transcends axioms. It's a truth about the natural numbers; not merely a truth about proofs in a formal system. — fishfry
I think that when you do math, you tend to be a Platonist; but when you try to defend the activity rationally, you have to fall back on being a fictionlist. — fishfry
I am more in agreement with @Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it.As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it. — Daz
The fact that you say "every photo, every dream..." does not entail that there is a finite number of photos or dreams.Yet the number of all those possible photos is not infinite. — Zelebg
How do you know this? What is the basis for this claim in your argument? — Cabbage Farmer
It seems to me that it's only what's called "the known universe" that is "established by scientific method".Depends on what you mean by "the universe". If the nature of the universe is established via the scientific method, whatever is the result must be finite. — Echarmion
It seems to me that it's only what's called "the known universe" that is "established by scientific method". — Cabbage Farmer
But there's an important conceptual difference between the world as it is, and the world as it is known by us.
I see no reason to suppose that our knowledge of the world at any given time in history would give us complete knowledge of the whole world.
Is there some reason to suppose that what we know about the universe at any given time, in keeping with scientific method, is all that we will ever come to know? — Cabbage Farmer
Is there some reason to suppose that the sum of everything we could ever possibly know about the universe, in keeping with scientific method, would provide a complete account of everything that is in fact the case, across all time and all space, or across whatever "dimensions" we should name alongside or instead of time and space, and across whatever universes and multiverses and iterations of generation and decay of universes or multiverses there may be....? — Cabbage Farmer
Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously. — aletheist
It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers." — aletheist
As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it.
— Daz
I am more in agreement with Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it. — aletheist
No worries, I always appreciate your point of view on philosophical aspects of mathematical matters.I should be careful. — fishfry
I am not telling anyone that they are doing something wrong. I suggest that philosophy is (among other things) about explaining what practitioners are actually doing, regardless of whether they accurately recognize it themselves. I have written extensively about philosophy of engineering, my own discipline, and colleagues find it fascinating because they never otherwise think about it in the way that I explain it; they just do it. For better or worse, most practitioners are not reflective practitioners in that sense.Philosophy is not about standing outside a given discipline and telling them they're doing it wrong. Philosophy has to be about explaining what practitioners are actually doing. — fishfry
Yes, in accordance with a certain set of definitions and axioms. Since we cannot point at a natural number to indicate what it is, all we have is a hypothesis from which we can and do draw necessary conclusions (like FLT).FLT is a statement about the natural numbers, and everybody knows exactly what they are. — fishfry
In Peircean terms, "abstract existence" is an oxymoron. Some abstractions are real, because they are as they are regardless of what any individual mind or finite group of minds thinks about them; but no abstractions exist, because they do not react with other things in the environment.Mathematical truths are abstract things, so they have abstract existence. — fishfry
Your claim "If the nature of the universe is established via the scientific method, whatever is the result must be finite", seems fair enough if it's a claim about the finitude of the current results of scientific method at any point in history, a claim about our knowledge.The known and the empirically knowable, yes. But beyond that, the meaning of "the universe" gets rather vague and nebulous. — Echarmion
Indeed. It seems to me these limitations are very much at issue here.This topic tends to run into language limitations. — Echarmion
Here again, it seems to me you've let your speech drag your claims and your beliefs beyond the bounds of evidence and reason.Well, yes, because by definition "what is in fact the case" is established by the scientific method. You probably mean that there might be large parts of reality forever hidden from any human mind. And that could be the case. Or it could not. But for practical purposes, it seems irrelevant. — Echarmion
Your claim "If the nature of the universe is established via the scientific method, whatever is the result must be finite", seems fair enough if it's a claim about the finitude of the current results of scientific method at any point in history, a claim about our knowledge. — Cabbage Farmer
Such a claim would resemble Zelegb's claim to have provided "proof that there is no infinity". Both claims purport to aim beyond what is empirically knowable. At most you can claim to show that our knowledge of the world is finite. But you cannot claim to show -- or how would you show? -- that our knowledge of the world gives us a perfectly complete account of the world as it is in fact.
By my account, those claims of yours and Zelegb's amount to speculation beyond the limits of empirical knowledge, and seem motivated by unwarranted conceptions of the relation of knowledge and reality. — Cabbage Farmer
By contrast, I have not claimed that the world is infinite. Rather, I say
(i) it seems we cannot know whether the world is finite or infinite in the relevant sense
(ii) surely the fact that our knowledge of the world is finite, or that "the world as we know it" is finite, is no proof that the world itself is finite
(iii) your claims seems to contradict both (i) and (ii). — Cabbage Farmer
Don't you agree that what is in the fact the case is in fact the case, whether or not we know it? Or do you suppose our knowledge creates reality in every regard? — Cabbage Farmer
Our knowledge of what is in fact the case is informed by experience and is made rigorous by scientific method. That does not entail that experience and scientific method establish what is the case and create or determine the whole world. — Cabbage Farmer
Wasn't this enough?We would never finish writing down the natural numbers. — TheMadFool
Infinity is something else. Somewhere, in the number pi, are all the phrases you have uttered during your life and, moreover, in the same order in which they were uttered. A little further on, there are all the books that disappeared because of the burning of the Library of Alexandria. In another place, there are all the speeches that Demosthenes gave and that he never wrote, but with the letters inverted, as in a mirror. Yes, the conception of what is infinite is too vast for me to grasp well in finite examples. — Borraz
This would actually be a weaker version of absolute normality - the property of containing every finite sequence of digits in every base with "equal frequency" (scare quotes because this is more complicated than it sounds). — SophistiCat
As far as I know, this has not been proven about any known number, — SophistiCat
↪fishfry Right, I was being sloppy, I must have had in mind computable numbers. Thanks. — SophistiCat
Specific speculative claims about what is unknowable are unwarranted conjectures.Yes, our knowledge, and therefore whatever model of reality is based on that knowledge, can only ever be finite. There might be unknowable aspects of reality, but given that they are unknowable, speculation on them is moot. — Echarmion
I'm not sure I would agree with this.My point was exactly that anything that is empirically knowable must be finite. — Echarmion
The claim we began by addressing is a claim to have proved that "there is no infinity". I take it you and I are still considering that claim when we use words like "universe" and "world" in this conversation.I further contend that "the universe" should refer to something empirical, as a matter of practicality. — Echarmion
I do not claim there is a "world behind the world". I say, by definition, there is one world; and it seems that world is knowable at least in part, on the basis of appearances.I just don't see why whether the "world behind the world" is or is not finite is "relevant". — Echarmion
In what way do you say our knowledge creates reality?I do not agree with that. I am a constructivist, so yes I do claim that, in a way, our knowledge creates reality. Not necessarily "in every regard" though, since I am not sure what you wish to imply with that. — Echarmion
Do you mean to say that experience and scientific method "populate the world with all the content" of the world? What does it mean to say this?They may not create the world, but they nevertheless populate it with all the content. All we can say about the world absent experience is that it exists. — Echarmion
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