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Fictionalism is an approach to theoretical matters in a given area which treats the claims in that area as being in some sense analogous to fictional claims: claims we do not literally accept at face value, but which we nevertheless think serve some useful function.

Thanks, bookmarked. One of the schools of mathematical nominalism is fictionalism.
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I did a short breakdown of the topic here:
That goes down a rabbit hole of info and posts to even more topics. Good reading.

I was looking at Steffan's slideshow, and it goes into how Cantor's axioms are paradoxical because the set of all sets has smaller cardinality that the set of all subsets of that set. But for similar reasons as have been discussed in this thread, I'm not convinced by it since the axioms only seem relevant to finite sets (similar to a sequence having a first and last step only being relevant to a finite set of steps), and none of the sets in the paradox is finite. So it's a bit like saying infinity squared is larger than infinity, which it isn't.
Far be it for humble me to not be as distressed by this as the hardened mathematicians. I take their word that this has more teeth than I see.

Also of interest is the mention of Godel demonstrating that a goal to find a complete and consistent foundation of mathematics cannot be reached. Does this mean that there cannot be one, or just that we cannot know it to be complete and consistent?

Yes, and I think that Lionino may have been protesting at such ways of talking. If one is not a platonist, the way to say what you want to say is to conceptualise "real" in a non-platonic way.
I have issues with what most people label 'realism', so I'm probably further from platonism than are most. Real is a relation to me, and I use the word that way.

I've noticed a variety of extensions of the use of "=" lately, so I'm sorry if I misused it.
OK, there can be more than one use of the symbol. We seem to not be in disagreement.

The modal fictionalism link is appreciated.
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I was looking at Steffan's slideshow

Mind you, an in-house mathematician has criticised some of the content in the slideshow (and in the article). Here is our exchange (I don't think he will mind me leaking DMs in this case):
Start with the first one: Cantor did not attempt to axiomatize mathematics. Cantor provided an understanding of mathematics in terms of sets, but he did not offer an axiomatization.
I got that statement off Vincent's slideshow slide 15 that I linked in the first paragraph "He did this by establishing set theory in an axiomatic way.". Is it wrong?
One might argue that informally implicit are the axiom schema of unrestricted comprehension and the axiom of extensionality; also the axiom of choice. But I don't know that Cantor articulated them as axioms.

Indeed, it is common in the basic literature to distinguish between, on the one hand, Cantor's work (sometimes called 'naive set theory') that was not formally axiomatized and, at best, deserving to be called 'an axiomatization' in only a overbroad sense and, on the other hand, actual axiomatizations such as those of Frege, Whitehead and Russell, and Zermelo.
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Not in those words. "Does not allow for a minute to pass", like somehow the way a thing is described has any effect at all on the actual thing.

Let me remind you, the "thing" being described here, in the op is a fictitious scenario. It is one hundred percent dependent on the description, just like a counterfactual. We might say that "the factual situation" is that a minute will pass, but the counterfactual described by the op does not allow for a minute to pass. You seem to be unable to provide the required separation between these two, thinking that the factual and the counterfactual may coexist in the same possible world.

Anyway, I see nothing in any of the supertask descriptions that in any way inhibits the passage of time (all assuming that time is something that passes of course).

Right, as I said there is nothing in the op to inhibit the passing of time, in fact the passing of time is an essential part, it is a constant. However, the premises of the op restrict the passing of time such that 60 seconds will not pass.

Ah, it slows, but never to zero. That's the difference between my wording and yours. Equally bunk of course. It isn't even meaningful to talk about the rate of time flow since there are no units for it. The OP makes zero mention of any alteration of the rate of flow of time.

There is nothing in the op to indicate that the passing of time slows. That is an incorrect interpretation. As you say, it isn't meaningful to talk about the rate of time in this scenario. What happens is that the speed of the person descending the staircase increases. And, as the speed increases, there is no limit to the acceleration indicated. The velocity is allowed to increase without limit. Even if we considered "infinite velocity" is a limited (which is of course contradictory), and assume that limit could be reached, this would still not imply "no time is passing". It would only make the spatial-temporal relationship unintelligible due to that contradiction.

This is actually very similar to the perspective of special relativity theory, which uses the speed of light as the limit, rather than infinite speed. This avoids contradiction but ti still renders the spatial-temporal relationship as unintelligible at the speed of light. From the perspective of the thing moving that fast, it appears like no time is passing, yet time is still passing. It's just a twisted way of making the passage of time relative to the moving thing for the sake of the theory. But there is no relativity theory stated in the op, nor any other frame of reference, so there is nothing to indicate a stopping, or even a slowing of time. The frame of reference which you keep referring to, in which 60 seconds passes, is excluded as incompatible with the described acceleration. The described acceleration is purely fictional though, like a counterfactual.
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But you just proved P2 yourself! You agreed that under the hypothesis of being able to recite a number at successively halved intervals of time, there is no number that is the first to not be recited.
— fishfry

I agreed that if P2 is true then C1 is true, as I have agreed from the beginning.

This doesn't prove that P2 is true.

You yourself proved P2 true, and I don't understand why you aren't even engaging with my argument supporting that claim.
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As if reality is the limit of our theories.
— fishfry
Since I don't know what "reality" means in its philosophical sense (which I designate by "Reality", but I do know, roughly, what you mean by "the limit of our theories", I would prefer to say "The limit of our theories is Reality". I'm of the school that teaches that the philosophical sense is metaphysics, and nonsense. But, since I arrived on these forums, I've had to recognize that, in philosophical discourse, "Reality" is a term in regular use and with some level of common understanding.

Reality is what's really going on in the world. Not sure why you regard that as problematic.

A bat has a particular view of the world, as does an ant, as does a sea slug. None of them, and that includes us, know what ultimate reality is. Not sure what your objection or concern is with this idea.

It's still a bit broad brush. I can understand it in the context of the inescapable inaccuracy of measurement in physics, etc, contrasted with the preternatural accuracy of (many, but not all) mathematical calculations. It's a version of Kant's regulative ideals and gives some content to phenomena/noumena and an explanation how they might be related.

Physics is inaccurate, but what if it's wildly inaccurate, as inaccurate as an ant's view of the world relative to the real world? We like to think that we're "close" to knowing reality because our physics works so well, but that's arguable.

Well, I would certainly want to get him to explain what he means by "is". That might slow him down a bit.

As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space.

Intellectuals have human motivations and follies just like everyone else - and some of them would do well to acknowledge that. I understand also that it is irresistibly tempting to explain people's failures to recognize conclusive rational arguments in ways that they will not like. But one needs also to understand that can be a trap. Hence Plato turned a classification of the philosophers he disagreed with into a term of abuse - "sophist", "rhetoric". You may have noticed that I'm engaged in some discussion with Metaphysician Undercover about this issue in relation to Zeno.

I have the worst habit lately of only responding to my mentions and not reading the rest of these threads.

They, and, apparently @noAxioms cannot believe that Zeno believed his own arguments - and that's not an irrational response because they are incredible. Nevertheless, I can't believe that they believe that. It's not easy. But I think it is important not to follow Plato's example in this respect.

Zeno's arguments are far more sensible than Tegmark's.

The example is so familiar to me that I thought it would add clarity. To the extent it got in the way, perhaps I should rethink how I present the idea.
— fishfry
I don't think there was anything wrong with your explanation. There's no such thing as the bullet-proof, instantly comprehensible, explanation. On the contrary, it helps to allow people space to turn what you say round and poke it and prod it. It's part of the process of coming to understand a new idea.

I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp.

The lamp's defined at each point of the sequence, but it's not defined at the limit.
— fishfry
Quite so. It's a sequence, but also a chain, because each point of the sequence depends on its predecessor. The reason it's not defined at the limit is that we can never follow the chain to its' conclusion - even thought the conclusion, the end, the limit, is defined.

But the limit isn't defined in the lamp problem. There is no limit to 0, 1, 0, 1, ..., and therefore no terminal state that's more natural than any other.

It seems paradoxical, because the limit is established before the chain can begin. The first step is to define the limit and the origin; that gives us something we can divide by 2 - and off we go.
This may not be mathematics. But I do maintain it is philosophy.

There need not be any relationship or rule that defines the elements of a sequence, but the sequence can have a limit. But in the case of the lamp, there's no natural conclusion because 0, 1, 0, 1, ... has no limit. I'm repeating myself but that's the point of the lamp. There's no natural terminal state.

The consequence is that the series "vanishes" if we try to look back from the "end". It's existence depends on our point of view. I don't suppose that any mathematician would be comfortable with that, but I plead that we are talking about infinity and standard rules don't apply.

I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply. It's vague, handwavy imaginings that don't apply.

I'm asking, in what sense? Surely math has never been fixed. It's always changing. It's a human activity.
— fishfry
Originated as, yes. But that doesn't restrict how math is seen today.
— fishfry
I think you are agreeing with me. Abstract today, applied tomorrow. Or often the reverse. We invent new abstract math to help us understand some real world application. It goes back and forth.
— fishfry
I agree with all of that. But I think it is very, even hideously, complicated.

Not following at all. Math is constantly changing, has been for thousands of years, and is changing even as we speak. New math papers are published every day.

Are you referring to some kind of Platonic math, God's math textbook? Is that what you mean by fixed?

It seems to me that we should always be specific about what is fixed and what is not. There may be disagreement about what goes in to which classification or what "fixed" means. But to say "math" without specifying further leads to confusion.

I don't feel confused, so I must not be understanding you.

Arithmetic, for example, is (relatively) fixed, though it may be modified from time to time. The inclusion of 0 and 1 as numbers is an example. Number theory might count as another example - I'm not sure about that. But once the methods of calculation are defined, they are fixed and the results from them are fixed as well.

Yes, sure, a fixed body of knowledge evolves. But that body of knowledge is added to every day by every math journal and university colloquium.

One could say, however, that both methods and results are discovered rather than defined, because there are ways of demonstrating whether a particular procedure gives the right result or not - through the application of the results or through the application of criteria like the consistency and completeness of the system.

Not only new results, but new ways of thinking are constantly introduced.

Euclidean geometry is similar, so far as I'm aware.
Algebra, calculus, non-Euclidean geometry, infinity theory are all additions to mathematics, rather than replacements of anything. It is almost irresistible to speak of them as developed or created rather than discovered, but since they share something with arithmetic and geometry, there are some grounds for speaking of them as discovered, because they were always possibilities, in some sense. What is it that is shared? The best I can do is to say something like logic - a sense of what is possible, or permitted.

So are you thinking of God's math book that humans slowly learn about? Well maybe that is fixed. So you're just regarding math as a Platonic body of knowledge that is "out there" somewhere. Is that what you mean? I'm not saying it's not true, I'm just asking if you're taking a Platonic viewpoint.

This is not irrelevant to this thread. Once we have realized that "+1"
can be applied to the result, it would not be wrong to say that the result of every step is fixed, whether or not we actually do add 1 to the 3,056th step. The result of each step is "always already" whatever it is. (I think it derives from Heidegger, but that doesn't prevent it from being helpful.) It captures the ambiguity between "+1" as something that we do and something that is done as soon as it is defined, or even before that.
As a result of the simple recognition of a possibility, we find ourselves plunged into a new and paradoxical world. I mean that it is simply not clear how the familiar rules are to be applied. Which makes it clear that we have to invent new ones - or are we discovering how the familiar rules apply or don't? I don't think there is a determinate answer and "always already" recognizes the ambiguity without resolving it.
When we refer to a step in the series, are we talking about something that we do (and may not do) and which actually takes time or something that is "always already" done, whether we actually ever do it or not?

I believe I lost track of what this paragraph referred to, sorry.
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You are assuming a non-realist view of mathematical entities again. You can still have Euclidean and non-Euclidean facts in the world as different facts just like algebra and calculus are different facts. Many philosophers think mathematical objects are real objects that exist outside of space and time.

The world can not be simultaneously Euclidean and non-Euclidean.

Algebra and calculus can both be true of the world. They don't contradict each other.

Not sure I'm following your analogy.

You can't have facts about the world that are in conflict with each other.

What does the realist say about that? I did a quick lookup of mathematical realism and it has nothing to do with the relationship of math to the world; rather, Google says, "Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is." The Wiki article on the philosophy of math takes a similarly abstract view of mathematical realism. Nothing to do with the physical world.

That's a lot more subtle than saying that realism believes that math is literally true in the world.

I make no claim to expertise on these matters, actually I admit to ignorance. But I don't think you are using mathematical realism in the same sense as Google and Wikipedia.
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You yourself proved P2 true

No I didn't.

Your argument is analogous to this:

If I am immortal then when will I die of old age? I won't. Therefore, I have proved that I am immortal.

Agreeing with what follows if we can recite the natural numbers at successively halved intervals of time doesn't prove that we can recite the natural numbers at successively halved intervals of time.
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I read your post. It is really helpful. I don't know enough to respond meaningfully, but I have a feeling I shall find my way back to it from time to time.

It did provoke the heretical speculation that it is an assumption that just one of these accounts applies to the whole of mathematics. Perhaps mathematics is not just one language-game, but a family of them.
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Reality is what's really going on in the world. Not sure why you regard that as problematic.
A bat has a particular view of the world, as does an ant, as does a sea slug. None of them, and that includes us, know what ultimate reality is. Not sure what your objection or concern is with this idea.
I agree with "a bat has.... what ultimate reality is" But then, I wonder what the status of "what's really going on in the world". Is that ultimate reality? From what you say, the answer is not clear. My concern is that both "ultimate reality" and "what's really going on in the world" are not defined in a way that reminds me of the way that the last step in a converging series is not defined - and cannot be defined. Yet, the sun is really shining at the moment and there really is a war in Ukraine - in short, we all (including bats and ants and slugs) live in the same world and interact in it.

Physics is inaccurate, but what if it's wildly inaccurate, as inaccurate as an ant's view of the world relative to the real world?
But how can you say that an ant's view of the world is inaccurate? I think I can grasp what you are getting at when you say that physics is inaccurate. It reflects the fact that physics is an on-going enterprise. "What if it's wildly inaccurate.." is a style of question that I'm very sceptical of. It reminds me of "what if everything's a simulation?" I classify it as a speculation and not capable of a meaningful answer.

As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space.
One might interpret that belief as a dramatic way of putting the point that we can find a mathematical structure that applies to the world. If he doesn't mean that, I want to know what he means by "is".

I have the worst habit lately of only responding to my mentions and not reading the rest of these threads.
A very sensible policy. It is easy to drive oneself crazy by trying to respond to everything. But sometimes I can't resist intervening in discussions that haven't mentioned me. It doesn't always work, in the sense of developing into something interesting, but some times it does.

I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp.
That's my fault. Sorry. I did benefit very much.

But the limit isn't defined in the lamp problem.
Yes, I understand that now. I was talking about the limit of the convergent series. The series "0,1,..." has no inherent limit. If it ever is limited, it is by some event "outside" the series. That's badly put. I just mean that I can stop following the instruction for any reason that seems good to me or even none at all. The series as defined is infinite.

I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply.
I didn't mean to suggest that wasn't the case. Thinking of the series backwards is a vague handwavy imagining. That's all. I intended to contrast that with a series that can be defined forwards or backwards. It's odd, that's all.

Yes, sure, a fixed body of knowledge evolves. But that body of knowledge is added to every day by every math journal and university colloquium.
Both sentences are true - the first sentence does not imply anything platonic, in my view. I think the difference between us is a question of emphasis rather than an actual disagreement.

I believe I lost track of what this paragraph referred to, sorry.
Yes, that was a step too far, and it is very speculative, more a musing than a thought. I should not have pursued it. Let's just let it go.
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The world can not be simultaneously Euclidean and non-Euclidean.

I am not talking about the fabric of space-time.

Nothing to do with the physical world.

Right, except for the kinds of realism that make it about the physical world, but that is one type among many.

Maybe you are misunderstanding what "abstract" means in those quotations. It doesn't mean something that we conceive in our minds, but a real object that exists independently of any conscious being, but that is outside space and time.

That's a lot more subtle than saying that realism believes that math is literally true in the world.

Of course a single sentence doesn't represent a family of views. But one of the minimal characteristics of mathematical realism is that things such as "2+2=4" are true and they are true even if we are all dead — in other words, it is about the world.

But I don't think you are using mathematical realism in the same sense as Google and Wikipedia.

I hope not, my sources are academic.
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Maybe you are misunderstanding what "abstract" means in those quotations. It doesn't mean something that we conceive in our minds, but a real object that exists independently of any conscious being, but that is outside space and time.
But one of the minimal characteristics of mathematical realism is that things such as "2+2=4" are true and they are true even if we are all dead — in other words, it is about the world.
If both of these are true, then we need to be very careful about what we mean by "the world". There is an application that takes "the world" to exist in space and time. Note, however, that the space-time world continues to exist even if we are all dead, even if we never existed at all. If "the world" includes everything that exists, then it can, of course, include things that exist "outside" of space and time - provided that we understand how anything can exist "outside" space, which seems to indicate a location, but does not.

Is not existing at any particular location in space the same as existing outside space? Where does platonism or Darwinism exist? or football or judo? Or the possibility of rain where I live tomorrow? Or the English language? Or the recipe for doughnuts?

Sorry - rhetorical questions. I realize that you are reporting the things that platonists might say.
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Agreeing with what follows if we can recite the natural numbers at successively halved intervals of time doesn't prove that we can recite the natural numbers at successively halved intervals of time.

Of course. Fully agreed. But that's YOUR hypothesis, not mine. Am I wrong about that? What am I missing here? Isn't that your example?

You have repeatedly asked me what happens if we go backwards, saying "1" at 60 seconds, "2" at 30 seconds, and so forth. That also is a purely hypothetical thought experiment. Why on earth are you proposing hypothetical non-physical thought experiments, then saying, "Oh that's impossible!" when I attempt to engage?
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I agree with "a bat has.... what ultimate reality is" But then, I wonder what the status of "what's really going on in the world". Is that ultimate reality? From what you say, the answer is not clear.

Don't know how to answer that. If there is an ultimate reality, whether we have access to it or not, that's what it is. If there isn't, then maybe it's all random. But then that would be the ultimate reality. God's point of view, as it were.

My concern is that both "ultimate reality" and "what's really going on in the world" are not defined in a way that reminds me of the way that the last step in a converging series is not defined - and cannot be defined. Yet, the sun is really shining at the moment and there really is a war in Ukraine - in short, we all (including bats and ants and slugs) live in the same world and interact in it.

No, those are only the things we perceive. Ants don't see the war but they do experience the sun. Creatures at the bottom of the sea don't see much of the sun. Ultimate reality is whatever is really out there, if there even is such a thing.

But how can you say that an ant's view of the world is inaccurate?

Limited is a better world. Our view is limited too.

I think I can grasp what you are getting at when you say that physics is inaccurate. It reflects the fact that physics is an on-going enterprise. "What if it's wildly inaccurate.." is a style of question that I'm very sceptical of. It reminds me of "what if everything's a simulation?" I classify it as a speculation and not capable of a meaningful answer.

Ultimate reality is the thing that physicists believe (or used to believe) they are trying to understand and learn about. That is no longer a core objective in physics, since the "shut up and calculate" school of quantum physics won the day.

As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space.
— fishfry
One might interpret that belief as a dramatic way of putting the point that we can find a mathematical structure that applies to the world. If he doesn't mean that, I want to know what he means by "is".

He says that the world literally is a mathematical structure. Not "is described by," but literally is.

https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

I have the worst habit lately of only responding to my mentions and not reading the rest of these threads.
— fishfry
A very sensible policy. It is easy to drive oneself crazy by trying to respond to everything. But sometimes I can't resist intervening in discussions that haven't mentioned me. It doesn't always work, in the sense of developing into something interesting, but some times it does.

I'm probably missing a lot.

I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp.
— fishfry
That's my fault. Sorry. I did benefit very much.

I love talking about the ordinals so if you enjoyed that I'm happy.

But the limit isn't defined in the lamp problem.
— fishfry
Yes, I understand that now. I was talking about the limit of the convergent series. The series "0,1,..." has no inherent limit. If it ever is limited, it is by some event "outside" the series. That's badly put. I just mean that I can stop following the instruction for any reason that seems good to me or even none at all. The series as defined is infinite.

It's infinite in the sense of being defined at each of 1, 2, 3, 4, ..., but it is NOT defined at $\omega$, the "point at infinity." Or for a more concrete representation, it's defined at each of 1/2, 3/4, 7/8, ..., but it is not defined at 1. That's the point. The 0, 1, 0, 1, ... sequence only covers the infinite sequence. But there is no definition of the state at the limit point.

I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply.
— fishfry
I didn't mean to suggest that wasn't the case. Thinking of the series backwards is a vague handwavy imagining.

That's @Micheal's example. But the sequence 1, 1/2, 1/4, 1/8, ... is a familiar sequence with limit 0. If you plot the points on the real line, it does indeed go backward, from right to left.

That's all. I intended to contrast that with a series that can be defined forwards or backwards. It's odd, that's all.

Not sure what a sequence (not series, a series is an infinite sum) that can be defined forwards or backwards means. By definition, an infinite sequence is $a_1, a_2, a_2, \dots$. It only goes forward. Though if the elements are decreasing (as 1, 1/2, 1/4, ...) the points go from right to left.

Yes, sure, a fixed body of knowledge evolves. But that body of knowledge is added to every day by every math journal and university colloquium.
— fishfry
Both sentences are true - the first sentence does not imply anything platonic, in my view. I think the difference between us is a question of emphasis rather than an actual disagreement.

Well, human endeavors are never fixed, they always evolve. In the case of math, it's a philosophical question as to whether there's something "out there" that it's evolving to.

I believe I lost track of what this paragraph referred to, sorry.
— fishfry
Yes, that was a step too far, and it is very speculative, more a musing than a thought. I should not have pursued it. Let's just let it go.

It looked interesting, I should take another look.
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The world can not be simultaneously Euclidean and non-Euclidean.
— fishfry

I am not talking about the fabric of space-time.

Perhaps I misunderstood. What then?

Nothing to do with the physical world.
— fishfry

Right, except for the kinds of realism that make it about the physical world, but that is one type among many.

Not conversant with the technical meaning of realism in this context, probably can't hold up my end of this.

Maybe you are misunderstanding what "abstract" means in those quotations. It doesn't mean something that we conceive in our minds, but a real object that exists independently of any conscious being, but that is outside space and time.

Is this a dualist point of view? What is outside of space and time?

I think the question, "Was 5 a prime number, before there was life in the universe?" is meaningless. Are you saying that a realist would say the answer is Yes? In that case, what is the difference between a realist and a Platonist?

I freely admit to my philosophical ignorance, so I am out of my depth in these matters.

Of course a single sentence doesn't represent a family of views. But one of the minimal characteristics of mathematical realism is that things such as "2+2=4" are true and they are true even if we are all dead — in other words, it is about the world.

But no, that is not about the world. The world is what's real, what's physical.

So you (or the realists) are arguing a Platonic position then. I don't happen to think 2 + 2 = 4 is true before there was intelligence in the universe. Or, say, before the big bang. Or before the endless cycle of big bangs, "big bounce" speculative cosmology. I think it takes a human, or at least an intelligence, to give meaning to the proposition.

If we are all dead, if there is no life and no intelligence in the world, who can pass judgment on whether 2 + 2 = 4? How could it be assigned a truth value? Who would do the assigning, the agreeing and the disagreeing?

Again, I admit my thoughts in this area are naive and not the product of any directed study of what the smart folks think about these matters. So I don't put a lot of stock in my own opinions.

Let me ask you a different question. Before chess was invented, did all the games of the grandmasters exist "out there" in Platonic space? Did the collected games of Bobby Fischer exist before he played them? After all, each game could be encoded as a number, and the Platonists believe numbers exist independently of minds. I find that difficult to believe, that all the symbolic works of humanity exited before they were created.

Humans create. That's what we do. Humans are, if you like, the very mechanism by which the universe figures out if 2 + 2 = 4.

I hope not, my sources are academic.

I have no doubt, and I hope I am sufficiently conveying the humble limits of my knowledge in this area.

But these academic sources, are they Platonists? Dualists? Surely there are those who strenuously disagree and take something approximating my own point of view. That 2 + 2 = 4 has no truth value before there is an intelligence to assign it one; and that the chess games of the grandmasters did not exist before they were played, even if the list of moves is an abstract sequence of symbols that a Platonist must believe existed before time itself.
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But that's YOUR hypothesis, not mine.

It's not mine. It's the hypothesis of those who claim that supertasks are possible. They try to use such things as the finite sum of a geometric series to resolve Zeno's paradox. They claim that because time is infinitely divisible it's possible for us to perform a succession of operations that correspond to a geometric series, and so it's possible to complete an infinite succession of operations in finite time.

I have been arguing firstly that it hasn't been proven that time is infinitely divisible and secondly that if we assume such a possibility then contradictions such as Thomson's lamp follow.

I was very clear on this in my reply to you on page 4, 22 days ago:

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction.

Most of the last few pages has been me trying to re-explain this to you, e.g. 10 days ago:

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed.

---

You have repeatedly asked me what happens if we go backwards, saying "1" at 60 seconds, "2" at 30 seconds, and so forth. That also is a purely hypothetical thought experiment. Why on earth are you proposing hypothetical non-physical thought experiments, then saying, "Oh that's impossible!" when I attempt to engage?

It was brought up for two reasons. The first was to address the flaw in your reasoning. That same post 10 days ago was very clear on this:

Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.

What natural number did I not recite?

...

Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not recite?

...

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed.

If argument 1 is proof that it is possible to have recited the natural numbers in ascending order then argument 2 is proof that it is possible to have recited the natural numbers in descending order.

It is impossible to have recited the natural numbers in descending order.

Therefore, argument 2 is not proof that it is possible to have recited the natural numbers in descending order.

Therefore, argument 1 is not proof that it is possible to have recited the natural numbers in ascending order.

---

The second reason I brought it up was a proof that it is impossible to have recited the natural numbers in ascending order.

If it is possible to have recited the natural numbers in ascending order then it is possible to have recorded this and then replay it in reverse. Replaying it in reverse is the same as reciting the natural numbers in descending order. Reciting the natural numbers in descending order is impossible. Therefore, it is impossible to have recited the natural numbers in ascending order.

Or if you don't like the specific example of a recording, then the metaphysical possibility of T-symmetry might suffice.

Either way, the point is that it's special pleading to argue that it's possible to have recited the natural numbers in ascending order but not possible to have recited them in descending order. It's either both or neither, and it can't be both, therefore it's neither.
• 932
I hope not, my sources are academic.
— Ludwig V
I have no doubt, and I hope I am sufficiently conveying the humble limits of my knowledge in this area.
There's a confusion here. The remark you quoted, which the system attributed to me, is actually @Lionino. I could claim academic sources from what I'm saying, but I read them a long time ago, and if you asked my for attributions, I would have to spend a long time looking them up.

By definition, an infinite sequence is a1,a2,a2,… It only goes forward. Though if the elements are decreasing (as 1, 1/2, 1/4, ...) the points go from right to left.
I take your point. So the dots reflect the lack of definition and trying to run it backward finds the dots at the "beginning", so the "beginning" is not defined. But one could define a similar sequence that runs (0, 1/2,1/4.... 1), couldn't one? That would not be the same sequence backwards, of course.

I freely admit to my philosophical ignorance, so I am out of my depth in these matters.
But no, that is not about the world. The world is what's real, what's physical.
Welcome to my world. Being out of one's depth in it is almost a prerequisite of inhabiting it, so that's not a problem. It would probably unfair to say that people who think they are not out of their depth are always wrong (compare relativity and QM). But it is certainly true that you need to be a bit out of your depth to be doing any serious work. If you have everything sorted out and pinned down, you've lost your grip on the problem. (Wittgenstein again)

Unfortunately "The world is what's real, what's physical" is a metaphysical remark (at least, it is if there are any philosophers around), so you've jumped into the water without, perhaps, intending to. The question is whether numbers, etc. are real things that are not physical; platonist-type theories see numbers as real things that "transcend" the physical world. Don't ask me what "transcend" means - or "thing", "entity", "object". They would probably prefer to tell you what transcendence etc. are. But that's the same question in a different mode. Their mode is metaphysics. Mine is linguistic.

What I was doing, in response to what Lionino was saying, was putting realism and anti-realism together - since they are defined in opposition to each other - and then asking what they disagree about. (There are many varieties of both sides of this coin, so I'm simplifying, and arguably distorting.) In particular, I'm trying to show that "real" is not 'really' in contention, since no-one could deny that numbers are real - what is at stake is different conceptions of reality. And you see how slippery this is because in mathematics, not only are some numbers real and some imaginary, other numbers (like transfinite ones) are neither. Worse still, the imaginary numbers are numbers and exist, so must be real - in the philosophical sense. (At least, you can put me right if I'm wrong here.)

What "real" means depends on the context in which you are using it. Some philosophers want to use "real" in a context-free sense. But that generates huge complications and confusion. Better to stick to contexts. (The same applies to "exists") That's why I try to avoid metaphysics and metaphysicians will classify me as a linguistic philosopher - and that is indeed where I learned philosophy.

Let me ask you a different question. Before chess was invented, did all the games of the grandmasters exist "out there" in Platonic space? Did the collected games of Bobby Fischer exist before he played them? After all, each game could be encoded as a number, and the Platonists believe numbers exist independently of minds. I find that difficult to believe, that all the symbolic works of humanity exi(s)ted before they were created.
All right. Those are good questions. They lead one in a certain direction. I am very sympathetic, so it would be better to let a platonist answer them directly. But I don't think that platonism needs to rule out the possibility that humans might be able to create some things, such as fictional stories - (although Plato was very scornful about such things on moral grounds, though he made liberal use of them himself.) - and games.
But in this field, it is as well to understand your opponent's (colleague, hopefully, in a joint attempt to discover truth) position. So consider. Games like chess are unlike games like football. Once they are defined, all the possible games are defined (so long as you limit the number of moves). So you could argue that the Sicilian defence, for example, was not created, but discovered. That's the germ of platonism.
In the end, I think, one has to see these arguments, not as simple question of truth and falsity, but of how you think about things. The answers, then, are quite likely to be pragmatic or even moral.

Humans create. That's what we do. Humans are, if you like, the very mechanism by which the universe figures out if 2 + 2 = 4.
Yes, that's fine. There is an approach that sees humans (and perhaps some animals) as the means by which the universe becomes self-conscious. I think that's going a bit too far, but I can see the attraction.
My enemy in this field is dogmatism.
• 14.4k
To make this very simple, we have two competing claims:

1. If we start reciting the natural numbers then either we stop on some finite number or we never stop

2. It is metaphysically possible to recite the natural numbers at successively halved intervals of time

If (2) is true then we can stop without stopping on some finite number.

Some take this as proof that (1) is false. I take this as proof that (2) is false.

I think that (1) is a tautology whereas no evidence has been offered in support of (2).
• 932
I think that (1) is a tautology
I agree. By "we" do you mean us human beings? You and I? If so, we will necessarily stop, if only when we die.

whereas no evidence has been offered in support of (2).
Assuming that there are people who believe this, it is reasonable to assume that they can offer what they think is evidence. So it's truth depends on what you mean by "evidence".
By "recite", do you mean some event that occupies a finite amount of time (larger than 0)? In that case, assuming you mean "all the natural numbers", 2 is false, or at least logically impossible.
• 3.6k
I think that (1) is a tautology whereas no evidence has been offered in support of (2)

What is "evidence" in a metaphysical realm?
• 12.5k
If (2) is true then we can stop without stopping on some finite number.

How do you make this conclusion?
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