And yet we can; and yet we do, map series of infinite numbers, one against the other. — Banno

In case you've never noticed this, claiming that you've dome something, and actually doing it, are two different things.

The only reason why no one can actually pair the integers is because they are stated to be infinite, and by this definition, it is impossible to do such. Therefore it is logically impossible to do such. — Metaphysician Undercover

Sorry, that is not how logical impossibility is defined. It would have to be something that is impossible for anyone even to conceive (like a square circle), not something that is merely impossible for anyone to do. Again, the latter is actual impossibility, which has absolutely no relevance whatsoever to pure mathematics.

What I have shown is that I cannot understand mathematics because the language of mathematics contradicts my native language, English. This renders mathematics as incoherent and unintelligible to me. — Metaphysician Undercover

My native language is also English, and I see no contradiction whatsoever with the language of mathematics that we have been discussing here. The same is true of any and every English-speaking mathematician in the world. In any case, it frankly seems rather foolish to keep making definitive (and incorrect) pronouncements about a subject that, by your own admission, you cannot even understand.

If you have a problem with my terms (they are English), then address my posts and tell me where the problems are. If my terms are not related to mathematics, then don't worry about them, they pose no threat to this field which you hold sacred. — Metaphysician Undercover

This is the exactly what I was talking about . The issue is you're misrepresenting what is being said. It should be patently obvious mathematicians do not define mapping (pairing) and infinity so as to make them jointly inapplicable. Just saying "I'm speaking English" isn't even beginning to honestly address this obvious fact. If your terms are not related to mathematics then you have absolutely no argument against the mathematical results relating to infinity. You're simply talking about something else.

As I said, it's a simple and clear case refusing to simply read how the terms are defined and then pretending to have discovered a problem because some colloquial definitions of some words conflict with the colloquial definitions of other words. Mathematics is formal, our definitions need to be stated up front/known beforehand and remain consistent throughout the calculation. If there is a contradiction, show the formal contradiction. Prove the system to be trivial.

It would have to be something that is impossible for anyone even to conceive (like a square circle), — aletheist

That's exactly my argument, it's logically impossible, impossible to conceive of, just like a square circle is logically impossible. The point is that people claim to be able to conceive of square circles, just like they claim to be able to conceive of pairing infinite numbers. People claim all sorts of weird things, like a polygon with infinite sides. They do this by violating, or changing the definitions of the terms.

This is the exactly what I was talking about . The issue is you're misrepresenting what is being said. It should be patently obvious mathematicians do not define mapping (pairing) and infinity so as to make them jointly inapplicable. Just saying "I'm speaking English" isn't even beginning to honestly address this obvious fact. — MindForged

That's the point. In English we know that pairing infinite numbers is impossible, just like we know that counting infinite numbers is impossible. The way that we use and define "infinite" and the way that we use and define "pairing", ensures that this is impossible. if mathematicians want to define these two terms in a different way, so that it is possible to pair an infinite number, that's their prerogative. I am not here to police mathematicians. However, we ought to be clear that this "mathematical" language is inconsistent with common English, and also inconsistent with how "infinite" is represented in philosophy.

If your terms are not related to mathematics then you have absolutely no argument against the mathematical results relating to infinity. You're simply talking about something else. — MindForged

You may have noticed that I have no arguments against the mathematical results relating to infinity, although others like Devans99 do. I really don't care about the mathematical results relating to infinity, because what "infinity" means to a mathematician is something completely different from what "infinity" means to me, a philosopher. And, I think it's quite obvious that the mathematicians have it wrong, (they've created an illusory "infinity"), so I'm really not interested in the conclusions which they might derive from their false premises.

That's exactly my argument, it's logically impossible, impossible to conceive of, just like a square circle is logically impossible. — Metaphysician Undercover

You have offered no argument for this claim, you have merely asserted it over and over; and now you have completely undermined your own position by freely acknowledging that you are not using the relevant terms in accordance with how they are carefully defined within mathematics, such that there is no logical impossibility whatsoever.

You have offered no argument for this claim, you have merely asserted it over and over — aletheist

Can you not read? Or do you have an extremely short memory? Let me reiterate. "Pairing", like "counting" is a human activity which is not successful unless it is completed. "Infinite" is defined in such a way that human activities such as counting and pairing cannot be completed an infinite number of times. Therefore it is logically impossible to count, or pair, an infinite number.

Of course I've spelled this out for you numerous times already, and you've simply ignored it, claiming some unreasonable distinction between logically impossible and actually impossible. So I expect you to continue with this unreasonable ploy.

now you have completely undermined your own position by freely acknowledging that you are not using the relevant terms in accordance with how they are carefully defined within mathematics, such that there is no logical impossibility whatsoever. — aletheist

This in no way undermines my position. I've acknowledged this from the beginning, defining "infinite" in a different way allows for infinite pairing. My position is that this mathematical definition of "infinite" is mistaken because it doesn't properly represent what "infinite" refers to in common usage, and in philosophy. The mathematical notion of "infinite" is illusory.

"Pairing", like "counting" is a human activity which is not successful unless it is completed. — Metaphysician Undercover

In mathematics, pairing and counting are not activities at all; they are concepts, and there is no requirement that they ever actually be completed, or even be capable of actually being completed. One more time: Mathematics has to do with the hypothetical, not the actual.

As I said, it's a simple and clear case refusing to simply read how the terms are defined and then pretending to have discovered a problem because some colloquial definitions of some words conflict with the colloquial definitions of other words. — MindForged

Exactly. No one is disputing that some mathematical definitions of terms are inconsistent with their colloquial (or even philosophical) meanings, but that has no bearing whatsoever on whether the associated concepts are logically possible.

In mathematics, pairing and counting are not activities at all; they are concepts, and there is no requirement that they ever actually be completed, or even be capable of actually being completed. One more time: Mathematics has to do with the hypothetical, not the actual. — aletheist

Right, so in mathematics you can count the infinite numbers without counting the infinite numbers.
That's exactly why I say it's a false premise. If you want to ignore that contradiction, and accept this hypothetical as a true premise that's your prerogative. I think this hypothetical is clearly and obviously false though, so I reject it as false, and I would not employ it as a premise, like mathematicians do.

What you refuse to acknowledge, is that I reject it on the basis that it is logically impossible by way of contradiction, because "infinite" means cannot be counted. You might claim that it's not logically impossible because "infinite" means something other than this in mathematics, but I think that's wrong, as an illusory definition of 'infinite".

That's my opinion and I will continue to defend it until someone demonstrates to me that I am wrong, that as something capable of being be paired or being counted is a better representation of what "infinite" truly represents. Simply pointing out that my opinion is inconsistent with the opinions of many mathematicians, does not demonstrate that my opinion is wrong.

In English we know that pairing infinite numbers is impossible, just like we know that counting infinite numbers is impossible. The way that we use and define "infinite" and the way that we use and define "pairing", ensures that this is impossible. if mathematicians want to define these two terms in a different way, so that it is possible to pair an infinite number, that's their prerogative. I am not here to police mathematicians. However, we ought to be clear that this "mathematical" language is inconsistent with common English, and also inconsistent with how "infinite" is represented in philosophy. — Metaphysician Undercover

You make a big deal out of an unimportant point. No one ought care about how these are defined in natural language because the meaning is often in flux, is context sensitive and still has multiple definitions. Sometimes we say "infinite" and mean Aristotle's potential infinity , sometimes we mean a completed infinity (as in the cardinality of an infinite set) and other times we just mean some arbitrarily large number that we leave unspecified. Philosophy always makes recourse to.mathematics in understanding infinity, I don't know why you think otherwise.

You may have noticed that I have no arguments against the mathematical results relating to infinity, although others like Devans99 do. I really don't care about the mathematical results relating to infinity, because what "infinity" means to a mathematician is something completely different from what "infinity" means to me, a philosopher. And, I think it's quite obvious that the mathematicians have it wrong, (they've created an illusory "infinity"), so I'm really not interested in the conclusions which they might derive from their false premises. — Metaphysician Undercover

How do you not see the contradiction between "I have no arguments against the mathematical results of infinity" and "I think it's quite obvious that the mathematicians have it wrong"? It's one or the other, either you're not arguing against it and thus you cannot say it's wrong, or else you're saying it's wrong and thus have some argument against it.

The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. Philosophers almost uniformly appeal to this rigorous understanding they gave us because it let's us come to grips with understanding this intuitive mathematical concept that we only vaguely understand in natural language. It caused many paradoxes in philosophical areas (e.g. Zeno's paradoxes) that were banished once mathematicians (not philosophers) gave a real regimentation of the concept. So of course we should privilege the mathematical understanding, which philosophers do. It's applicable to many areas of philosophy, mathematics, science, you name it. And it still accords with many intuitions about infinity, though not all of them (which doesn't matter since the intuitive understanding of infinity creates paradoxes).

The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. — MindForged

I would add Peirce here, although as in the case of philosophy, unfortunately his contributions are widely overlooked.

That's fair. I don't know much about Pierce outside vague recollections of semiotics as an undergrad, and a somewhat obscure fact that he discovered/created classical logic at almost the same time as Frege did (although even Frege didn't get the credit initially, despite Russell emphatically crediting him for it).

Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity.

Sometimes we say "infinite" and mean Aristotle's potential infinity , sometimes we mean a completed infinity (as in the cardinality of an infinite set) and other times we just mean some arbitrarily large number that we leave unspecified. Philosophy always makes recourse to.mathematics in understanding infinity, I don't know why you think otherwise. — MindForged

The fact that philosophy has a different definition of infinite which is inconsistent with your mathematical definition of "completed infinity" is clear evidence that philosophy does not make recourse to mathematics for its understanding of reality.

The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. Philosophers almost uniformly appeal to this rigorous understanding they gave us because... — MindForged

All this demonstrates is that you are very selective in the philosophy which you read. Cantor's representation of "infinite" was confronted by Russell, and hence replaced by Zermelo-Fraenkel. But any thorough reading on the subject will reveal that the issue is far from settled.

Nevertheless there was, and still is, serious philosophical opposition to actually infinite sets and to ZF's treatment of the continuum, and this has spawned the programs of constructivism, intuitionism, finitism and ultrafinitism, all of whose advocates have philosophical objections to actual infinities. Even though there is much to be said in favor of replacing a murky concept with a clearer, technical concept, there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for. — Internet Encyclopedia of Philosophy

Notice specifically, "..there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for". This is my argument. By redefining "infinite" mathematics is not even dealing with what we generally refer to as "infinite'. It has created a completely new concept of "infinite". It has put aside the true concept of "infinite" which derives its meaning from continuity, in favour of an illusory one, a completed one, in order to create the illusion that it has resolved the problems of infinity. In reality the new concept of "infinite" has just distracted us from the true infinite. "Completed" is not a word which one could use to describe "infinite" in the way that we commonly use the word. Of course, your claim is that the "completed infinity" is the true concept of infinity. I disagree.

How do you not see the contradiction between "I have no arguments against the mathematical results of infinity" and "I think it's quite obvious that the mathematicians have it wrong"? It's one or the other, either you're not arguing against it and thus you cannot say it's wrong, or else you're saying it's wrong and thus have some argument against it. — MindForged

It's as I said, I have no arguments against the conclusions drawn by mathematicians from their concept of "infinite", what you call the "results". I do not even know these conclusions, or results, and I have no interest in them. I am arguing against their premise, their concept of "infinite". This is not contradictory, just a simple statement of fact, I am not arguing against the results (conclusions), I am arguing against the premise (their concept of "infinite"). And, I have no interest in these results.

Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity. — aletheist

Right, tell that to MindForged, who seems to think that mathematicians have resolved the philosophical problem of "infinity". In reality, mathematicians have redefined "infinity" to suit their own purposes, neglecting the real problem of infinity, which is associated with continuity. And this might lead some naïve philosophers to think that mathematicians have resolved the problem of infinity. All they've really done is created a new problem, a divided concept of "infinity".

The fact that philosophy has a different definition of infinite which is inconsistent with your mathematical definition of "completed infinity" is clear evidence that philosophy does not make recourse to mathematics for its understanding of reality. — Metaphysician Undercover

Philosophy does not have a different definition of infinity outside the colloquial ones which are inconsistent. A potential infinity is just that: potential, as in not an infinity. Check out any relevant excerpts from mereological and ontological work that relate to infinity and in virtually all of them infinity is understood based on the long-standing mathematical definition of it. The reasons for this should be obvious.

All this demonstrates is that you are very selective in the philosophy which you read. Cantor's representation of "infinite" was confronted by Russell, and hence replaced by Zermelo-Fraenkel. But any thorough reading on the subject will reveal that the issue is far from settled. — Metaphysician Undercover

There was opposition to Cantor early on, to be sure. But that was gone in relatively short order. The Cantorian understanding of infinity is the understanding of infinity in modern mathematics, any claim that ZFC is markedly different from Cantor is just false. It's completely settled in current mathematics. In fact, check the paragraph immediately before the one you quoted from the IEP:

Finally, by the mid-20th century, it had become clear that, despite the existence of competing set theories, Zermelo-Fraenkel’s set theory (ZF) was the best way or the least radical way to revise set theory in order to avoid all the known paradoxes and problems while at the same time preserving enough of our intuitive ideas about sets that it deserved to be called a set theory, and at this time most mathematicians would have agreed that the continuum had been given a proper basis in ZF.
[...]
Because of this success, and because it was clear enough that the concept of infinity used in ZF does not lead to contradictions, and because it seemed so evident how to use the concept in other areas of mathematics and science where the term “infinity” was being used, the definition of the concept of "infinite set" within ZF was claimed by many philosophers to be the paradigm example of how to provide a precise and fruitful definition of a philosophically significant concept. Much less attention was then paid to critics who had complained that we can never use the word “infinity” coherently because infinity is ineffable or inherently paradoxical.

Notice specifically, "..there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for". This is my argument. By redefining "infinite" mathematics is not even dealing with what we generally refer to as "infinite'. It has created a completely new concept of "infinite". It has put aside the true concept of "infinite" which derives its meaning from continuity, in favour of an illusory one, a completed one, in order to create the illusion that it has resolved the problems of infinity. I — Metaphysician Undercover

All you're proving is as I said, that some colloquial definitions conflict with others. Who cares? If those definitions lead to insoluble paradoxes and cannot be applied where they ought to (in mathematics) then they need replacing. With the proper understanding of infinity and a developed calculus, we solves Zeno's paradoxes when philosophers could not because they did not have a workable definition of infinity outside the vaguely defined one. Mathematics does not wholecloth redefine infinity, it still has most of the properties it intuitively ought to have (continuously extendable, for instance), but has the unique benefit of being perfectly and probably consistent.

And it's funny that you mention constructivism and such from the IEP. The problem is - ignoring that constructivism does not eliminate all infinities - is that those are in the extreme minority, even in philosophy. Worse, the quote you mentioned is talking about the continuum, not infinity in general. Thats the size of the real numbers, not of the set of natural numbers which is still completed in intuitionistic mathematics (constructivism). Ultrafinitism is widely regarded as just above crankery, funnily enough because they argue similarly to you that we should reject infinity and essentially pretend that the counter-intuitive properties of infinity should be treated as if they are contradictions even though it's not even arguable because we have formal proofs that infinity does not introduce any inconsistencies in standard mathematics.

It's as I said, I have no arguments against the conclusions drawn by mathematicians from their concept of "infinite", what you call the "results". I do not even know these conclusions, or results, and I have no interest in them. I am arguing against their premise, their concept of "infinite". This is not contradictory, just a simple statement of fact, I am not arguing against the results (conclusions), I am arguing against the premise (their concept of "infinite"). And, I have no interest in these results. — Metaphysician Undercover

You haven't argued against it save to say that it's different than the colloquial one in *some* respects. So if the definitions cannot be agreed upon, we need only look at the results. Your view of infinity neuters mathematics because then calculus goes out the window as that requires several sizes of infinity (you're dealing with the real numbers, for one) and science since it is predicated on ZFC and uses calculus everywhere (not to mention all current spacetime theories that aren't mostly speculative (e.g. Lopp-quantum gravity) explicitly assume space and time are a continuum). That's just a useless definition at that point, especially as it then runs counter to other colloquial views on infinity.

Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity.
— aletheist

Right, tell that to MindForged, who seems to think that mathematicians have resolved the philosophical problem of "infinity". In reality, mathematicians have redefined "infinity" to suit their own purposes, neglecting the real problem of infinity, which is associated with continuity. And this might lead some naïve philosophers to think that mathematicians have resolved the problem of infinity. All they've really done is created a new problem, a divided concept of "infinity". — Metaphysician Undercover

Peirce was writing in the exact time that Cantor and Dedekind's work on infinity was contentious. It's just dishonest to pretend that has any bearing in the status of that work among philosophers and mathematicians today.

Philosophy does not have a different definition of infinity outside the colloquial ones which are inconsistent. A potential infinity is just that: potential, as in not an infinity. Check out any relevant excerpts from mereological and ontological work that relate to infinity and in virtually all of them infinity is understood based on the long-standing mathematical definition of it. The reasons for this should be obvious. — MindForged

I am not talking about potential vs. actual. I am talking about "infinity" as boundless (philosophical conception), and "infinity" as completed (mathematical conception). The two are incompatible.

All you're proving is as I said, that some colloquial definitions conflict with others. Who cares? If those definitions lead to insoluble paradoxes and cannot be applied where they ought to (in mathematics) then they need replacing. With the proper understanding of infinity and a developed calculus, we solves Zeno's paradoxes when philosophers could not because they did not have a workable definition of infinity outside the vaguely defined one. Mathematics does not wholecloth redefine infinity, it still has most of the properties it intuitively ought to have (continuously extendable, for instance), but has the unique benefit of being perfectly and probably consistent. — MindForged

Choosing one conception and rejecting the other does not resolve the incompatibility. Nor does it resolve the paradoxes involved with the one conception, by choosing the other conception. That's simply an act of ignorance.

if you could demonstrate that "infinity" (the philosophical concept) as boundless, and incomplete, is an incoherent, unintelligible conception, then we'd have reason to reject it in favour of the other, mathematical conception. Until then it remains a valid concept which is incompatible with the mathematical concept of "infinitely".

On the other hand, I reject the mathematical conception because I believe it was created solely for the purpose of giving the illusion that the issues involved with the philosophical concept of "infinite", as boundless and incomplete, could be resolved in this way, by replacing the conception. Despite your claims about how calculus and science rely on this conception of "infinite", I believe it serves no purpose other than to create the illusion that the problems involved with the philosophical concept of "infinity" have been resolved. In reality, mathematics could get along fine without this conception of "infinity". It would just be different, having different axioms. And, since this conception of infinity is just a distraction for mathematics, mathematics would probably be better without it.

Peirce was writing in the exact time that Cantor and Dedekind's work on infinity was contentious. It's just dishonest to pretend that has any bearing in the status of that work among philosophers and mathematicians today. — MindForged

Not sure if this was directed at me, but to clarify--I did not mean to imply that Peirce's objections to Cantor and Dedekind carry much weight among philosophers and mathematicians today. They do not, which I happen to think is unfortunate, but only from the standpoint of understanding true continuity as utterly incompatible with discrete mathematics. I readily acknowledge that treating the real numbers as if they constituted a continuum works just fine for most practical purposes within mathematics.

Nah, it was directed at the other guy.

I am not talking about potential vs. actual. I am talking about "infinity" as boundless (philosophical conception), and "infinity" as completed (mathematical conception). The two are incompatible — Metaphysician Undercover

That's not a philosophical conception, that's as much the colloquial conception as anything else. The two aren't incompatible either since in a sense even standard math has infinity as boundless. After all, take some arbitrary infinite set and new members can be added to it.

Choosing one conception and rejecting the other does not resolve the incompatibility. Nor does it resolve the paradoxes involved with the one conception, by choosing the other conception. That's simply an act of ignorance. — Metaphysician Undercover

Yes it does. These are competing theories of what infinity is and this it is not a debate that it fundamentally any different than any other disagreement in philosophy regarding how to define or conceptualize something. Zeno's paradoxes are not resolvable under the colloquial understanding of infinity, but they are resolved by appeal to modern mathematics (calculus) which requires the hierarchy of infinities. It's not an act of ignorance, it's then use of a better theory of infinity because it's both usable in mathematics and it resolves issues that existed previously. Under your view absolutely nothing can ever replace a previous misconception because to change ones accepted theory of a concept entails just changing the subject.

On the other hand, I reject the mathematical conception because I believe it was created solely for the purpose of giving the illusion that the issues involved with the philosophical concept of "infinite", as boundless and incomplete, could be resolved in this way, by replacing the conception. Despite your claims about how calculus and science rely on this conception of "infinite", I believe it serves no purpose other than to create the illusion that the problems involved with the philosophical concept of "infinity" have been resolved. In reality, mathematics could get along fine without this conception of "infinity". It would just be different, having different axioms. And, since this conception of infinity is just a distraction for mathematics, mathematics would probably be better without it. — Metaphysician Undercover

This is like rejecting nominalism because one thinks it was created to give the illusory victory of overcoming issues with platonism or something. In reality, mathematics cannot get by without the conception of infinity it uses. Otherwise you're doing something like constructivist mathematics which is more limited, using different tpea of analysis (e.g. smooth infinitesimal analysis and the like) and is significantly more limited in the proofs that can be made since Excluded Middle cannot be placed inside the universal quantifier. Virtually all science uses the ZFC set theory which includes the axiom of infinity. If the math all worked out without that axiom it would not be asserted as an axiom. You're essentially supposing all mathematicians are idiots who don't realize they have an unneeded or useless axiom despite the many criticisms of the formalism (including Cantor's work on infinity) of a century ago.

The smallest amount of time physics can "make sense of" is planck time.

So, yes, in a way, time is discrete.

Another way to look at it is to consider nerve thresholds. There's a limit to our senses. If something moves fast all we see is a blur. Fast enough and it disappears. Think of a bullet.

So, there is a lower limit of time we can experience and "make sense of".

But you're talking about a more fundamental property of time; that it doesn't flow but actually hops from moment to moment.

One problem with your argument is how you use NOW.

NOW is a moment in time. If it is 7:00 AM NOW then it means SEVEN hours have passed since midnight. NOW is just an arbitrarily chosen DISTANCE from a given starting point. It doesn't have a size I agree but it isn't meant to.

So, yes, NOW=0.

Suppose you're given a length of time 2 seconds.

How many NOWs are there in it?

We do the division x=2 seconds ÷ NOW. As NOW approaches zero (the real NOW=0) x approaches infinity. That is the correct answer because between any two points in a continuous model of time there should be infinite points or NOWS.

That's not a philosophical conception, that's as much the colloquial conception as anything else. — MindForged

Are you familiar with Platonic dialectics? We determine the meaning of a word by referring to how it is used in our society. This mean that the colloquial conception is the correct one. If mathematics is using a conception of "infinite" which is inconsistent with the colloquial conception, then this is an indication that they have not properly represented "infinite"?

Zeno's paradoxes are not resolvable under the colloquial understanding of infinity, but they are resolved by appeal to modern mathematics (calculus) which requires the hierarchy of infinities. — MindForged

Zeno's paradoxes were adequately resolved by Aristotle's distinction between actual and potential. The colloquial conception of "inifinite" is consistent with this distinction, though the colloquial understanding does not all the time include an understanding of this distinction, so Zeno's paradoxes may appear to one who holds the colloquial conception but does not understand Aristotle's resolution. The modern world of scientific discovery has long ago rejected Aristotelian physics, and with it the Aristotelian distinction between actual and potential. The concept of energy is clear evidence that this distinction has been lost to modern science. Because these principles, which resolve Zeno's paradoxes, were lost to modern science, Zeno's paradoxes reappeared as valid paradoxes.

The principles of modern mathematics do not resolve Zeno's paradoxes because the philosophers of mathematics have simply produced an illusory conception of "infinite", which is inconsistent with what we are referring to in colloquial use of the term. That's sophistry, and Platonic dialectics was developed as a means to root out and expose such sophistry. The sophists would define a word like "virtue" in a way which suited their purposes, and then profess to be teachers of this. However, Socrates exposed that what they were teaching as "virtue" was just their own little conception, which was completely inconsistent with the colloquial meaning of "virtue" (what the members of society in general regarded as virtue). Philosophers of mathematics have engaged in the same form of sophistry. They teach their own private conception of "infinite" which is completely inconsistent with what we generally mean by "infinite" (the colloquial meaning of the word), creating the illusion that this resolves Zeno's paradoxes.

Under your view absolutely nothing can ever replace a previous misconception because to change ones accepted theory of a concept entails just changing the subject. — MindForged

This is not true. What I am arguing is that if we change the defining features of a thing, then we are not talking about the same thing any more. Therefore we ought to give it a different name so as to avoid confusion. This is not a case of correcting a misconception, it is a case of introducing a new conception. We cannot say that one is a correction of a misconception, because they are distinct conceptions, having distinct defining features. The new conception ought to be named by a word which will not cause confusion with the old conception, or any sort of equivocation. For example, if the defining feature of parallel lines is that they will never meet, and someone says that they've come up with a new geometry in which parallel lines meet, then we ought not call these lines parallel, but use a term other than "parallel" in order to avoid confusion and the appearance of contradiction. They are distinct conceptions, not a correction of a misconception. Likewise, the new conception in mathematics, which is called "infinite" ought to bear another name like "transfinite" so as not to confuse the conception with what we commonly call "infinite".

You're essentially supposing all mathematicians are idiots who don't realize they have an unneeded or useless axiom despite the many criticisms of the formalism (including Cantor's work on infinity) of a century ago. — MindForged

That's ridiculous. I am saying no such thing, and I resent that because I have great respect for mathematicians, they are as far from "idiot" as you can get. But the mathematicians which I know do not create the axioms, as this is more of an activity of philosophical speculation. And I do believe that much of the philosophical speculations which provide the foundation for modern science and mathematics is misguided. And I would not call a misguided philosopher an idiot, because much philosophy is hit and miss, trial and error.

We do the division x=2 seconds ÷ NOW. As NOW approaches zero (the real NOW=0) x approaches infinity. That is the correct answer because between any two points in a continuous model of time there should be infinite points or NOWS. — TheMadFool

Alternatively, as Peirce argued, there are no instants (NOWs) in any continuous interval of time, and there are no points in any continuous segment of a line. Time does not consist of instants and space does not consist of points; instead, those are arbitrary discontinuities that we mark within continuous space-time for our own purposes, such as measurement. The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum.

Alternatively, as Peirce argued, there are no instants (NOWs) in any continuous interval of time, and there are no points in any continuous segment of a line. Time does not consist of instants and space does not consist of points; instead, those are arbitrary discontinuities that we mark within continuous space-time for our own purposes, such as measurement. The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum. — aletheist

I agree with this representation, but the problem which TheMadFool points to is that there appears to be real points of discontinuity within time, which are represented by the concept of Planck time. So if time is represented as a true continuity, as you say Peirce suggests, how do we account for these fundamental units which cannot be further divided. What type of point would mark the beginning and end of these units of time?

I think the question is whether the Planck time is properly described as a discrete "unit of time" or as a limitation on our ability to mark and measure time, which in itself is truly continuous. Needless to say, I lean toward the latter.

The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum. — aletheist

So, what can model a true continuum?

I'm a bit confused here. I consider the real numbers to be an adequate model of any continuum for we can measure an arbitrarily small or large quantity without any problem.

I agree that the real numbers are adequate for modeling, marking, and measuring discrete quantities, no matter how small. My understanding is that conceptualizing a true continuum requires the acceptance of infinitesimals. Again, for Peirce this was primarily a philosophical matter--related to his self-described "extreme scholastic realism" regarding generals--rather than a mathematical one.

My understanding is that conceptualizing a true continuum requires the acceptance of infinitesimals — aletheist

I agree. And infinitesimals are just 1/∞. And actual infinity is not a number:

1. If actual infinity is a number, there must be a number larger than any given number.
2. But that’s contradictory.
3. Can’t be a number AND larger than any number.
4. So actual infinity is not a number
5. Invention of magic numbers runs contrary to the spirit of science.

If actual infinity is not a number, a mathematical continuum does not exist IMO. Likely neither in reality...

No one is arguing for actual infinity, or that infinity is a number, or (for that matter) that infinitesimals are numbers. Again, numbers are intrinsically discrete, and thus useful for marking and measuring. On the other hand, switching from mathematics to philosophy, a real general is a true continuum--between any two actual instantiations, there are potential instantiations beyond all multitude.

But we have larges amounts of evidence supporting the statement 'reality is modelled by maths'. If sound maths cannot represent a continuum, then thats strong evidence against continuums existing.

there are potential instantiations beyond all multitude — aletheist

So those instantiations are not multitudes. So the actual instantiations are not multitudes. So you can't compare the two.

But we have larges amounts of evidence supporting the statement 'reality is modelled by maths'. If sound maths cannot represent a continuum, then thats strong evidence against continuums existing. — Devans99

Mathematical modeling is representation for a particular purpose. I already acknowledged that the real numbers serve as an adequate model of a continuum for almost all uses within mathematics, even though they do not themselves constitute a true continuum. Likewise, the mathematical models that I routinely create as part of my job are extremely useful within my professional field of structural engineering, even though they obviously are not actual structures being subjected to gravity, wind, earthquakes, etc.

And again, no one is arguing that true continua exist or are actual, but that they are real--they are as they are, regardless of what anyone thinks about them. Thus defined, reality is not coextensive with existence/actuality--there are also real qualities/possibilities and real habits/laws (conditional necessities); but that obviously takes us far beyond the thread topic.

Continua are real in our minds only. Along with infinity and talking trees. Anything is real in our mind; maths and reality are very different in my view; they are constrained to contain logical concepts only.