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. — Stanford Encyclopedia - Modal Fictions
That goes down a rabbit hole of info and posts to even more topics. Good reading.I did a short breakdown of the topic here: — Lionino
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.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. — Ludwig V
OK, there can be more than one use of the symbol. We seem to not be in disagreement.I've noticed a variety of extensions of the use of "=" lately, so I'm sorry if I misused it.
I was looking at Steffan's slideshow — noAxioms
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.
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. — noAxioms
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). — noAxioms
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. — noAxioms
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. — Michael
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. — Ludwig V
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. — Ludwig V
Well, I would certainly want to get him to explain what he means by "is". That might slow him down a bit. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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? — Ludwig V
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. — Lionino
You yourself proved P2 true — fishfry
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.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. — fishfry
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.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? — 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".As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space. — 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 have the worst habit lately of only responding to my mentions and not reading the rest of these threads. — fishfry
That's my fault. Sorry. I did benefit very much.I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp. — 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.But the limit isn't defined in the lamp problem. — fishfry
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.I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply. — 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.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
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.I believe I lost track of what this paragraph referred to, sorry. — fishfry
The world can not be simultaneously Euclidean and non-Euclidean. — fishfry
Nothing to do with the physical world. — fishfry
That's a lot more subtle than saying that realism believes that math is literally true in the world. — fishfry
But I don't think you are using mathematical realism in the same sense as Google and Wikipedia. — fishfry
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. — Lionino
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.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. — Lionino
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. — Michael
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. — Ludwig V
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. — Ludwig V
But how can you say that an ant's view of the world is inaccurate? — Ludwig V
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. — Ludwig V
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". — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
That's all. I intended to contrast that with a series that can be defined forwards or backwards. It's odd, that's all. — Ludwig V
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. — Ludwig V
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. — Ludwig V
The world can not be simultaneously Euclidean and non-Euclidean.
— fishfry
I am not talking about the fabric of space-time. — Lionino
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. — Ludwig V
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. — Ludwig V
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. — Ludwig V
I hope not, my sources are academic. — Ludwig V
But that's YOUR hypothesis, not mine. — fishfry
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. — Michael
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. — Michael
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? — fishfry
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. — Michael
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.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. — fishfry
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.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. — fishfry
I freely admit to my philosophical ignorance, so I am out of my depth in these matters. — fishfry
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)But no, that is not about the world. The world is what's real, what's physical. — fishfry
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.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. — fishfry
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.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. — fishfry
I agree. By "we" do you mean us human beings? You and I? If so, we will necessarily stop, if only when we die.I think that (1) is a tautology — Michael
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".whereas no evidence has been offered in support of (2). — Michael
If (2) is true then we can stop without stopping on some finite number. — Michael
How do you make this conclusion? — Metaphysician Undercover
If I recite the first number after 30 seconds, the second after 15 seconds, and so on, then I have recited them all and so stopped after 60 seconds, even though there is no largest number for me to stop on. — Michael
No, it isn't the same as being stopped. Being stopped is an everyday occurrence. Infinite speed, is, as you say, unintelligible. If that's what underpins the supertasks, it makes sense of the narratives - apart from the fact that it doesn't answer the question whether the lamp is on or off.I have to disagree. What you describe is a rate of acceleration which would produce an infinite speed. The rate at which you recite the numbers becomes infinite before 60 seconds passes. And, despite the fact that infinite speed is in some sense unintelligible, it is clearly not at all the same as being stopped. — Metaphysician Undercover
No, it isn't the same as being stopped. Being stopped is an everyday occurrence. Infinite speed, is, as you say, unintelligible. If that's what underpins the supertasks, it makes sense of the narratives - apart from the fact that it doesn't answer the question whether the lamp is on or off. — Ludwig V
If one watches the lamp in a dark room, at some point it will appear to be on continuously. — jgill
But that's YOUR hypothesis, not mine.
— fishfry
It's not mine. It's the hypothesis of those who claim that supertasks are possible. — Michael
They try to use such things as the finite sum of a geometric series to resolve Zeno's paradox. — Michael
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. — Michael
I have been arguing firstly that it hasn't been proven that time is infinitely divisible — Michael
and secondly that if we assume such a possibility then contradictions such as Thomson's lamp follow. — Michael
I was very clear on this in my reply to you on page 4, 22 days ago: — Michael
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.
— Michael — Michael
Most of the last few pages has been me trying to re-explain this to you, e.g. 10 days ago: — Michael
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.
— Michael — Michael
It was brought up for two reasons. The first was to address the flaw in your reasoning. — Michael
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? — Michael
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? — Michael
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.
— Michael — Michael
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. — Michael
It is impossible to have recited the natural numbers in descending order. — Michael
Therefore, argument 2 is not proof that it is possible to have recited the natural numbers in descending order. — Michael
Therefore, argument 1 is not proof that it is possible to have recited the natural numbers in ascending order. — Michael
The second reason I brought it up was a proof that it is impossible to have recited the natural numbers in ascending order. — Michael
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. — Michael
Replaying it in reverse is the same as reciting the natural numbers in descending order. — Michael
Reciting the natural numbers in descending order is impossible. — Michael
Therefore, it is impossible to have recited the natural numbers in ascending order. — Michael
Or if you don't like the specific example of a recording, then the metaphysical possibility of T-symmetry might suffice. — Michael
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. — Michael
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. — Ludwig V
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.
— fishfry
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. — Ludwig V
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) — Ludwig V
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. — Ludwig V
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.) — Ludwig V
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. — Ludwig V
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. — Ludwig V
My enemy in this field is dogmatism. — Ludwig V
Quite so. I've been thinking that the rules of the game require one to classify that as a purely physical phenomenon. But I prefer versions of this problem that define a sequence (0,1,0,1,...) and align that with the lamp. Even better, I think, we can count the steps in the convergent series and not that odd and even numbers alternate and ask whether the last count when 60 seconds are up is odd or even. Nothing physical intervenes.If one watches the lamp in a dark room, at some point it will appear to be on continuously. — jgill
That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light.I think that if the lamp is going on and off at an infinite rate, then it's not correct to say that it would be on at any particular time, or off at any particular time, because it is going on and off at a rate faster than our ability to determine a particular time. — Metaphysician Undercover
I'm not clear whether you are thinking of reciting as a human action that takes time. In which case, there will come a point in your recitation when you physically have to stop, but have not run out of natural numbers. (If we are talking about a series that is convergent in time, it will take longer to utter the word(s) than the time available.)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. — Michael
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