Continua are real in our minds only. — Devans99

But that is a contradictory statement--"real" means "as it is regardless of what anyone thinks about it," so anything that is "in our minds only" cannot be real.

so anything that is "in our minds only" cannot be real — aletheist

No I mean some things can exist in our mind and not in reality or mathematics. Illogical things like inanimate objects that talk. Or infinity. Or a true continua.

But by definition, if it exists in our minds only, it does not exist anywhere else. We conventionally class the contents of our minds as not part of reality. But you could take the opposite definition I guess.

No I mean some things can exist in our mind and not in reality — Devans99

Right, we call such things fictions. Unicorns are not real because they are as they are only because people think of them that way.

Illogical things like inanimate objects that talk. Or infinity. Or a true continua. — Devans99

There is nothing illogical about infinity or true continua, and the fact that they cannot exist (be actual) does not entail that they cannot be real.

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"? — Metaphysician Undercover

The problem is the word is not used in one single way. The way you prefer is inconsistent at times and at others only vaguely understood and has limits in both application to mathematics and in just straightforward analysis (e.g. understanding the divisibility of time and space). The notion of a definition being "incorrect" because it diverges from colloquial usage is absurd. It captures most or all of the features of that usage but without any contradictions at all.

Zeno's paradoxes were adequately resolved by Aristotle's distinction between actual and potential. — Metaphysician Undercover

This is a claim there is no reason to accept. Aristotle believed actual infinities were impossible but they are not. So the entire justification for his distinction between which infinities were possible is empty this side of Cantor. You cannot study a variable which does not exist within a fixed domain. For there to be a potential infinity, there has to be a predefined set of values that can be occupied otherwise the domain is precluded from study as it can change arbitrarily by an arbitrary amount. That domain is actually infinite. And so to understand that domain you need to understand and apply the mathematical regimentation of the concept of infinity.

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. — Metaphysician Undercover

This is just an outright misrepresentation. Mathematicians did not simply redefine infinity to mean something contrary to its colloquial usage to disingenuously prove things about it. And hold this thought, I'll come back to it later when you say something inconsistent with the above.

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". — Metaphysician Undercover

There is so much wrong here that it shows a deep lack of understanding of mathematics. "Defining features" are, ironically by definition, established by the definition in use. Otherwise we would never had words whose meaning varies across context and circumstance due to the resemblance in those varying contexts. Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). The actual Euclidean definition of what a parallel line is does not say the lines will not intersect. It's that if you have some infinite Line J, and a point P not on that line, no lines passing through P intersect with L. This does not hold if the space is different. It's only when one misstates the Parallel Postulate that it sounds contradictory to have intersecting parallel lines. You are forgetting that these notions are defined by the geometry, not separate from them.

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 — Metaphysician Undercover

So you say this, but remember what you said earlier?:

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. — Metaphysician Undercover

So you're not calling them idiots, you have respect for them, but you're calling them (or at least certainly comparing them to) sophists? Ok man. Alternative views of math are fine but you're muddying the waters by suggesting the idea is "illusory" simply because you don't think it completely preserves the intuitive idea of infinity (which is contradictory and fails to solve paradoxes). If a paradox arises in some domain of discourse then something has to change to resolve the paradox. Aristotle unwittingly required assuming the existence of the actual infinite in order to deploy the idea of a potential infinity (because the possibility space must be predefined to have any meaning, and indefinite segments are only possible in an infinitely divisible or extensive space), and the intuitive notion of an actual infinity is inconsistent.

So we looked to the mathematicians (Cantor, Bolzano, Dedekind and co.) who gave a concrete, comprehensible theory of the infinite that removed all contradiction while retaining most of the natural ideas about infinity. That's called success, not illusion and sophistry.

But I just showed that infinity is not a number. It definitely does not play by logical rules (see Hilbert's Hotel and all the other paradoxes of infinity). Nature on the other hand does play by logical rules. No place for magic in nature so no place for infinity either.

Maths describes reality to a high degree... no infinity in maths suggest no infinity in nature.

Have you read up on the hyperreals before? It's not anything I've studied but it looks like it's intended to capture this "true continuum" you're talking about if its comparison to infinitesimals means what I think it means.

Aristotle believed actual infinities were impossible but they are not — MindForged

They are so!

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.

"Actual infinity" is a word. The cardinality of the set of natural numbers is actually infinite and a number, namely the number aleph-null. That number is not larger than any given number (aleph-1, the size of the continuum, is larger), it's just larger than any natural number. Deploying a deductive argument is kind of silly when you make mistakes from the very beginning.

But I just showed that infinity is not a number. It definitely does not play by logical rules (see Hilbert's Hotel and all the other paradoxes of infinity). Nature on the other hand does play by logical rules. No place for magic in nature so no place for infinity either.

Maths describes reality to a high degree... no infinity in maths suggest no infinity in nature. — Devans99

Do you understand the point of Hilbert's Hotel? David Hilbert was a mathematician, not someone who rejected the notion of infinity as contradictory. The point of the "paradox" (not an actual paradox, just a strange thought experiment) is to point out that infinity is weird and does not work the way most people could naturally understand without learning some of the mathematical logic underlying our theories of infinity.

I think you have not addressed my argument... you have just introduced magic numbers that obey different, mad rules to all other numbers.

I did address your argument. Your first premise was simply false (hierarchy of infinities). I didn't introduce any magic numbers, and the theory of cardinalities of sets is the same for finite and infinite numbers.

hierarchy of infinities — MindForged

Infinity is defined to be bigger than anything else. That means there can only be one infinity by definition.

Infinity is defined to be bigger than anything else. That means there can only be one infinity by definition. — Devans99

No it isn't. Infinity in mathematics simply means some set's members can be put into a one-to-one correspondence with a proper subset of the parent set. What falls out of this is there can be multiple sizes of infinity. Infinity in math is not "The biggest number" or whatever. Aleph-null is larger than any natural number certainly, but aleph-null is smaller than the size of the continuum. Cantor proved this fact with a proof by contradiction (the Diagonal Argument).

Sigh. I give up. You are very set in your ways.

Have you read up on the hyperreals before? — MindForged

I suspect that I have encountered them, but it was a while back and I never got very far. I am aware of a few different approaches that seek to capture a true continuum mathematically, such as nonstandard analysis and smooth infinitesimal analysis. Again, I concede that the real numbers are an adequate model for almost all mathematical purposes.

The cardinality of the set of natural numbers is actually infinite — MindForged

Like Peirce, I prefer to say that it is really infinite, but not actually infinite. I also join Peirce in denying that numbers exist--i.e., I am not a mathematical Platonist--even while affirming that they are real.

The point of the "paradox" (not an actual paradox, just a strange thought experiment) is to point out that infinity is weird and does not work the way most people could naturally understand without learning some of the mathematical logic underlying our theories of infinity. — MindForged

Indeed, and that is the precisely point that you and I have been trying to make throughout this thread, albeit from somewhat different perspectives. Trying to treat infinity/continuity no differently from finite/discrete quantities is what leads to severe misunderstandings.

Aleph-null is larger than any natural number certainly, but aleph-null is smaller than the size of the continuum. Cantor proved this fact with a proof by contradiction (the Diagonal Argument). — MindForged

Right, and Peirce proved that the power of the set of all subsets of a given set is always greater than the power of the original set itself--which entails that there is no largest multitude (his term for aleph). What he called a true continuum is "supermultitudinous," larger than any multitude, and thus impossible to construct from (or divide into) discrete elements. You might find this introductory article about "Peirce's Place in Mathematics" interesting.

Again, I concede that the real numbers are an adequate model for almost all mathematical purposes. — aletheist

Oh I wasn't asking you to concede anything, I just started googling for some stuff and came across it. :) it looks like an attempt to recapture infinitesimals so it caught my eye.

Like Peirce, I prefer to say that it is really infinite, but not actually infinite. I also join Peirce in denying that numbers exist--i.e., I am not a mathematical Platonist--even while affirming that they are real. — aletheist

Oh that's fine, I'm just using the terminology I saw others using. I'm not sure I'm a math platonist either (undecided). I just meant it's a real one (in the sense that it's not just some continuously iterated task that still comes out to a finite number at every step).

Right, and Peirce proved that the power of the set of all subsets of a given set is always greater than the power of the original set itself--which entails that there is no largest multitude (his term for aleph). What he called a true continuum is "supermultitudinous," larger than any multitude, and thus impossible to construct from (or divide into) discrete elements. You might find this introductory article about "Peirce's Place in Mathematics" interesting. — aletheist

I'll take a look at it! From your description it sounds like what Cantor referred to as the Absolute Infinite though it doesn't exist in ZF.

Unicorns are not real because they are as they are only because people think of them that way... — aletheist

...which is what makes them real! They are not real in the same way that rocks and crocodiles are real. They're real like Harry Potter and mathematics are. For unicorns, Harry and mathematics are all human inventions that have no existence in the space-time universe that science so ably describes. They were invented for quite different reasons, admittedly, but none of them are "real" as you use the term. :up:

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. — aletheist

As I understand the issue, the physical evidence indicates that there is likely a real limit the spatial-temporal existence at this level. If this is the case then there is a separation, a lack of correspondence, between the conceptualization of a continuous space and time, and real physical existence. I believe that Peirce suggested replacing "infinite" with "infinitesimal", in our conceptions, as a way to deal with this problem.

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. — TheMadFool

It is impossible that any numbers can model a true continuum because all conceptions of numbers are based in a conception of units, such that a number signifies either a unit or a multitude of units, and this is incompatible with a true continuum. This issue is very similar to the issue with true infinity. The "infinity" is defined as something which numbers cannot count, it is impossible by definition, that numbers can count infinity. Likewise, it is impossible by definition that numbers can represent a true continuum.

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

Infinitesimals does not resolve the problem because infinitesimals are units. So to model a continuum as infinitesimals is to model it as composed of discrete units.

It captures most or all of the features of that usage but without any contradictions at all. — MindForged

Obviously, this is what I disagree with. The mathematical conception of "infinite" clearly contradicts the colloquial definition of "infinite", I've demonstrated this over and over again, so you know what I mean and I will not demonstrate it here again. You simply assert that it does not contradict, while the evidence is clear, that it does.

Neither of the two distinct conceptions have contradiction inherent within. There is no contradiction inherent within the colloquial concept. Where the problem lies is in applying the concepts to what we consider as the real world. Each conception, the colloquial and the mathematical, has its own set of problems involved with application.

This is a claim there is no reason to accept. Aristotle believed actual infinities were impossible but they are not. — MindForged

That's not a proper representation of what Aristotle argued. He used the argument to separate "eternal" from "infinite" because Ideas, Forms, were described as eternal, and "infinite" was an idea. So he proceeded to demonstrate that "eternal" and "infinite" were incompatible. What he demonstrated is that anything eternal is necessarily actual, while anything infinite has the nature of potential. The latter, that the infinite belongs in the class of potential, must be read as a definition, a description, derived from observation. All instances of "infinite" are conceptual, ideas, and ideas are classed in the category of potential. From this premise, along with other premises, the conclusion that anything that is eternal is necessarily actual is derived.

This idea which you state, that actual infinities are possible, is produced from the conflation of Aristotle's two distinct aspects of reality, actual and potential, and consequentially time and space, in the modern conception of "energy". So the claim "actual infinities are possible" (which would be contradictory if we adhered to Aristotle's distinction between actual and possible), only demonstrates a failure to maintain Aristotle's principles. it doesn't mean that "actual infinities are possible" has any coherent meaning.

. For there to be a potential infinity, there has to be a predefined set of values that can be occupied otherwise the domain is precluded from study as it can change arbitrarily by an arbitrary amount. — MindForged

The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to. To assign a set of values, in order to study that domain is simple contradiction.

That domain is actually infinite. — MindForged

See, you have taken the category which is defined by "that which cannot be studied", "potential", which consists of matter, ideas, and the infinite, and you've applied some values (which is contradictory), and now you claim that this thing "infinite" is no longer in that category, it's in the category of actual. All you have done is changed the subject.

This is just an outright misrepresentation. Mathematicians did not simply redefine infinity to mean something contrary to its colloquial usage to disingenuously prove things about it. And hold this thought, I'll come back to it later when you say something inconsistent with the above. — MindForged

Yes it does, read the above.

"Defining features" are, ironically by definition, established by the definition in use. Otherwise we would never had words whose meaning varies across context and circumstance due to the resemblance in those varying contexts. — MindForged

This is clearly false, and indicates a misunderstanding of how logic and understanding proceeds. We identify a thing (law of identity), this thing as identified, becomes our subject, and we proceed to understand it through predication. The "defining features", how the subject is defined, ensures that the subject represents the object. This is known as correspondence, truth. It is evident therefore, that "defining features" is determined by correspondence between the logical subject and the object which is said to correspond to that subject, and not by "the definition in use". If it were the "definition in use" which defined the subject everything would be random with no correspondence to reality It is clear that "the definition in use" must be consistent with the known correspondence, truth. When the definition in use is not consistent to provide a correspondence with the identified object, we can correct the definition, saying that you are using an incorrect definition.

Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). The actual Euclidean definition of what a parallel line is does not say the lines will not intersect. It's that if you have some infinite Line J, and a point P not on that line, no lines passing through P intersect with L. This does not hold if the space is different. It's only when one misstates the Parallel Postulate that it sounds contradictory to have intersecting parallel lines. You are forgetting that these notions are defined by the geometry, not separate from them. — MindForged

So this is very wrong because you have reified space, as if "space" were the subject, and there is a corresponding object which has been identified as "space". There is no such object being described here in geometry. The objects are all mathematical, conceptual, such as a "line". My point was that if there are two distinct concepts of "line", then there are two distinct objects referred to by that name "line" corresponding to the defining features which constitute the two distinct subjects under that name, "line". Therefore "line" ought not be used for identification of both of these objects.

Reality and existence are not synonyms. Harry Potter and unicorns are not real, because their properties depend upon the thinking of an individual mind (e.g., J. K. Rowling) or a finite group of minds. Rocks and crocodiles are real, because their properties do not have any such dependence.

How to characterize mathematics is not as straightforward. I endorse Charles Sanders Peirce's definition of it, which he adapted from that of his father Benjamin, one of the foremost mathematicians of the 19th century: the science which draws necessary conclusions about hypothetical states of things. This clearly rules out mathematical Platonism, which claims that objects such as numbers exist; they are (logical) possibilities, rather than actualities. Nevertheless, it may still be the case that those objects are real, in the sense that we discover their properties, rather than inventing them.

Infinitesimals does not resolve the problem because infinitesimals are units. So to model a continuum as infinitesimals is to model it as composed of discrete units. — Metaphysician Undercover

There are two fundamental mistakes here: first, infinitesimals are not units; second, a continuum is not composed of infinitesimals.

What we know, study, and understand, are all forms and forms are by definition actualities. — Metaphysician Undercover

Forms are only actualities when and where they are instantiated in concrete particulars. In themselves, as distinct from Matter, they are real possibilities.

Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". — Metaphysician Undercover

It violates the laws of classical (bivalent) logic, but that is not the only kind of logic available to us. For example, we can reason without the law of excluded middle using intuitionist logic. In fact, the law of excluded middle does not apply to infinitesimals; rather than discrete points, they are analogous to indefinite "neighborhoods" with an inexhaustible supply of potential points.

There are two fundamental mistakes here: first, infinitesimals are not units; second, a continuum is not composed of infinitesimals. — aletheist

Yes, infinitesimals are units, they are bounded necessarily in order to give them the status of distinct things "infinitesimals". without separation between them, there could not be the plurality indicated by "infinitesimals". Under Peirce's philosophy though, the boundaries are vague. However, it is necessary that either the boundaries are real in order that the infinitesimals are real, or else the boundaries are not real, in which case neither are the infinitesimals.

It violates the laws of classical (bivalent) logic, but that is not the only kind of logic available to us. For example, we can reason without the law of excluded middle using intuitionist logic. In fact, the law of excluded middle does not apply to infinitesimals; rather than discrete points, they are analogous to indefinite "neighborhoods" with an inexhaustible supply of potential points. — aletheist

Right, but doing this only gives "potential" a different definition, just like mathematics gives "infinite" a different definition. Producing a different definition only means that the corresponding object is not the same object. So we are no longer talking about what Aristotle identified as "potential" we are talking about something different.

Yes, infinitesimals are units, they are bounded necessarily in order to give them the status of distinct things "infinitesimals". — Metaphysician Undercover

No, infinitesimals are not units, and they are not "distinct things."

However, it is necessary that either the boundaries are real in order that the infinitesimals are real, or else the boundaries are not real, in which case neither are the infinitesimals. — Metaphysician Undercover

False, infinitesimals are real but indefinite--i.e., potential not actual.

Right, but doing this only gives "potential" a different definition, just like mathematics gives "infinite" a different definition. — Metaphysician Undercover

How so? By "potential" I simply mean real possibility, rather than actuality. As Peirce put it, "the word 'potential' means indeterminate yet capable of determination in any special case" (CP 6.185; 1898, emphasis in original).

No, infinitesimals are not units, and they are not "distinct things." — aletheist

You seem to have little understanding of what "infinitesimals" refers to. An infinitesimal is defined by Leibniz as an entity, a unity. It was used as an approach to mathematizing space, time, and matter which were previously conceived of as continuities. There was an inconsistency between mathematics which deals with distinct units, and these concepts, space, time, and matter, which were based in an assumption of continuity. If these continuities, space, time, and matter, could be conceived of as composed of infinitesimals (units, like monads) we could establish compatibility between a continuum and the numbers..

Peirce did nothing to change the fundamental nature of infinitesimals as units. The problem of course, is that if the things which we knew of as continuous, space, time, and matter, are really composed of these units, infinitesimals, then they are not truly continuous. So the question is whether these basic, primitive intuitions which hold space, time, and matter as continuous are correct, or are these things which appear to be continuous, more appropriately represented by the discrete units, infinitesimals. If the latter is the truth then we ought to be able to distinguish real boundaries between one infinitesimal and another. It does not suffice to say that the boundaries are there, but they are vague and cannot be determined. To give reality to the infinitesimals we need to determine those boundaries.

You seem to have little understanding of what "infinitesimals" refers to. — Metaphysician Undercover

Right back at you.

If these continuities, space, time, and matter, could be conceived of as composed of infinitesimals (units, like monads) we could establish compatibility between a continuum and the numbers. — Metaphysician Undercover

As I already stated, a continuum is not composed of infinitesimals. Moreover, there is no ultimate compatibility between a continuum and the numbers--or anything else discrete.

So the question is whether these basic, primitive intuitions which hold space, time, and matter as continuous are correct, or are these things which appear to be continuous, more appropriately represented by the discrete units, infinitesimals. — Metaphysician Undercover

The only reason for positing infinitesimals (in contrast to points) is to preserve those basic, primitive intuitions of continuity, rather than resorting to (wrongly) treating a continuum as if it were composed of discrete units.

Moreover, there is no ultimate compatibility between a continuum and the numbers--or anything else discrete — aletheist

That's the point, they are incompatible. And, the introduction of infinitesimals does not make them compatible. It's only an illusion.

The only reason for positing infinitesimals (in contrast to points) is to preserve those basic, primitive intuitions of continuity, rather than resorting to (wrongly) treating a continuum as if it were composed of discrete units. — aletheist

The problem with points is that a point is by definition, dimensionless. It takes up no space on a line, only divides a line. Therefore the line cannot be made up of points. Even an infinity of dimensionless points cannot account for the existence of the continuous, one dimensional line, which is supposed to exist between two points. There is a fundamental incompatibility between the point which is dimensionless, and the line, which is dimensional. Infinitesimal points were introduced to allow that a multitude of infinitesimals may have dimensionality, therefore a continuous line could be conceived of as being composed of infinitesimal points. This is an attempt to establish compatibility between the non-dimensional point, and the dimensional line.

Likewise, it is impossible by definition that numbers can represent a true continuum — Metaphysician Undercover

Continuum to me implies a smooth transition between two points. I agree that, numerically, we need to define a unit before we can make sense of a given quantity.

However, we can choose arbitrarily small units and do any measurement. What I mean is we can make any quanitification arbitrarily smooth depending on our needs for accuracy or whatever.

Also, the real numbers includes ALL points possible. In other words no point on the number line is left out. Isn't that sufficient for a continuum model?

Obviously, this is what I disagree with. The mathematical conception of "infinite" clearly contradicts the colloquial definition of "infinite", I've demonstrated this over and over again, so you know what I mean and I will not demonstrate it here again. You simply assert that it does not contradict, while the evidence is clear, that it does. — Metaphysician Undercover

You misread what you quoted. I said the mathematical conception has no contradictions, I didn't say it was identical to the colloquial definition. The colloquial understanding of infinity includes, for example, a notion of unboundedness. And yet we know infinities are in some sense bounded, and people will readily admit that the real between 0 and 1 are infinite despite that clearly being a bounded array of values. That's just an obvious case of a colloquial, folk conception being contradictory and hence the need for a formal understanding which we got from mathematics.

What he demonstrated is that anything eternal is necessarily actual, while anything infinite has the nature of potential. The latter, that the infinite belongs in the class of potential, must be read as a definition, a description, derived from observation. All instances of "infinite" are conceptual, ideas, and ideas are classed in the category of potential. From this premise, along with other premises, the conclusion that anything that is eternal is necessarily actual is derived. — Metaphysician Undercover

Oh my Lord, you aren't saying anything different. This is essentially just that an actual infinite isn't possible, but now because of our observations instead of an inherent contradiction. We don't derive from observation that the infinite is relegated to ideas. All current theories of spacetime that are more grounded than speculative (e.g. LQG isnt mainstream right now) require space and time to be infinitely divisible and no observation contradicts this at all. In fact, attempting to make those finite will result in inconsistencies more than likely. So no, observation does not require one to class the infinite as merely potential, a mere idea that cannot be found in the world.

And in any case, the way you're talking about potentiality sounds contradictory. It's being spoken of as if it's ineffable. And yet you're telling me about it and what makes it ineffable... Which means youre talking about it, so it's not ineffable.

The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to. To assign a set of values, in order to study that domain is simple contradiction. — Metaphysician Undercover

This sounds incredibly wacky. For one, even if there is some metaphysical violation of Excluded Middle, that doesn't preclude it from human understanding nor does that make reality "violate the laws of logic" because there are not "the" laws of logic. There are many such sets of laws, and some drop Excluded Middle. It would certainly be a surprise to the Intuitionists that they don't understand constructive mathematics or their own logic because it doesn't assume EM as an axiom.

See, you have taken the category which is defined by "that which cannot be studied", "potential", which consists of matter, ideas, and the infinite, and you've applied some values (which is contradictory), and now you claim that this thing "infinite" is no longer in that category, it's in the category of actual. All you have done is changed the subject. — Metaphysician Undercover

Ok so now you admit what you earlier rejected? Previously I said that under your view, a previous misconception of disagreement about a concept cannot changed because to do so is to change the subject. In which case progress is impossible because people cannot have different theories about the same concept. There goes all of philosophy.

If the category cannot be studied how do you know anything about it? If you don't know anything about it, how can it be studied? If you do know something about it you must be studying it by some means. In which case the distinctions you're making don't seem motivated by anything other than philosophical prejudice.

We identify a thing (law of identity), this thing as identified, becomes our subject, and we proceed to understand it through predication. The "defining features", how the subject is defined, ensures that the subject represents the object. This is known as correspondence, truth. It is evident therefore, that "defining features" is determined by correspondence between the logical subject and the object which is said to correspond to that subject, and not by "the definition in use". If it were the "definition in use" which defined the subject everything would be random with no correspondence to reality It is clear that "the definition in use" must be consistent with the known correspondence, truth. When the definition in use is not consistent to provide a correspondence with the identified object, we can correct the definition, saying that you are using an incorrect definition. — Metaphysician Undercover

What a highly idealized and incoherent procedure you're suggesting. The definition in use communicates how we believe the object to be. Predication is how we understand the object to have the attributes it has, it's not how we understand the object itself (that's an intuitive project, one done reflexively most of the time). This doesn't leave it random. If I say the term "Beep" refers to my dog, so long as my use is consistent absolutely everyone understands what I'm talking about (if they speak English). That's why people can make up words and have them often times be understood by people not perceiving what we are referring to.

And this is all besides the point anyway. The nonsense you tried to pass off earlier was the idea that there are "defining features" of things like parallel lines despite now knowing that these terms are defined by the user's (implicitly or explicitly). They don't have inherent definitions, they're defined within a certain domain. So the idea that you tried to push that parallel lines don't intersect is purely based off the underlying assumptions of the geometry in which you made an assumption of, it is not true writ large in geometry. The Parallel Postulate is only true inasmuch as it's assumed to be so in a geometry and anyone saying otherwise is just misinformed about how mathematical formalisms work.

So this is very wrong because you have reified space, as if "space" were the subject, and there is a corresponding object which has been identified as "space". There is no such object being described here in geometry. The objects are all mathematical, conceptual, such as a "line". My point was that if there are two distinct concepts of "line", then there are two distinct objects referred to by that name "line" corresponding to the defining features which constitute the two distinct subjects under that name, "line". Therefore "line" ought not be used for identification of both of these objects. — Metaphysician Undercover

I haven't reified anything, I didn't treat these as anything other than abstract mathematical constructs. Are you truly unable to talk about the basic properties of a geometry? There are different kinds of space indifferent geometries. Some are curved, some aren't as they are planes. Terms like "line" are usually left as undefined primitives in geometry so there is no confusion here because they are essentially take to be the same object in a different background (a different space) or else as a similar objects in different spaces so there's no benefit to calling one a line and calling nother "line-ish".

Continuum to me implies a smooth transition between two points. — TheMadFool

Continuum is uninterrupted. "Two points" implies two distinct places and therefore a boundary which separates them.

However, we can choose arbitrarily small units and do any measurement. What I mean is we can make any quanitification arbitrarily smooth depending on our needs for accuracy or whatever. — TheMadFool

So I assume that the "smooth transition" you refer to is arbitrary and not real?

You misread what you quoted. I said the mathematical conception has no contradictions, I didn't say it was identical to the colloquial definition. The colloquial understanding of infinity includes, for example, a notion of unboundedness. And yet we know infinities are in some sense bounded, and people will readily admit that the real between 0 and 1 are infinite despite that clearly being a bounded array of values. That's just an obvious case of a colloquial, folk conception being contradictory and hence the need for a formal understanding which we got from mathematics. — MindForged

I agree that the mathematical conception has no inherent contradiction. But the only sense in which "an infinity" is bounded is by the terms of its definition. All infinites which we speak of are bounded by the context in which the word is used. If someone mentions an infinity of a particular item, then the infinity is bounded, defined as consisting of only this item. Likewise if we are talking about an infinity of real numbers between 0 and 1, the infinity is bounded, limited by those terms. However, we are not discussing particular infinities here, which may be understood as particular (though imaginary) objects, we are discussing the concept of "infinite". We are not discussing conceptual entities which are said to be infinite, we are discussing what it means to be infinite. The reals between 0 and 1 is a conceptual entity which is said to be infinite. We are asking what does it mean when we say that this is infinite.

We don't derive from observation that the infinite is relegated to ideas — MindForged

This is false. Anytime "infinite" is used to refer to something boundless, or endless, it refers to something made up by the mind, something imaginary or conceptual. We do not ever observe with our senses anything which is boundless or endless, because the capacities of our senses are limited and could not observe such a thing. Since the capacities of our senses are finite we know that anything which is said to be infinite is a creation of our minds, it is conceptual, ideal.

All current theories of spacetime that are more grounded than speculative (e.g. LQG isnt mainstream right now) require space and time to be infinitely divisible and no observation contradicts this at all. In fact, attempting to make those finite will result in inconsistencies more than likely. So no, observation does not require one to class the infinite as merely potential, a mere idea that cannot be found in the world. — MindForged

Spacetime is conceptual. This is the problem I had with your last post, you reified "space", making it into some sort of an object to justify your position. In reality, "space" is purely conceptual. We do not sense space at all, anywhere, it is a constructed concept which helps us to understand the world we live in. Furthermore, "infinitely divisible" is an imaginary activity, purely conceptual. We never observe anything being infinitely divided, we simply assume, in our minds, that something has the potential to be thus divided.

And in any case, the way you're talking about potentiality sounds contradictory. It's being spoken of as if it's ineffable. And yet you're telling me about it and what makes it ineffable... Which means youre talking about it, so it's not ineffable. — MindForged

I never defined "potentiality" as ineffable. It may appear to you that potentiality is contradictory ifyou do not understand the concept, but Aristotle was very specific and explicit in his description of what the term refers to, to ensure that his conception does not defy the law of non-contradiction. As that which may or may not be, he allowed potentiality to defy the law of excluded middle. You will find that some modern philosophies though, such as dialectical materialism, and dialetheism allow for violation of the law of non-contradiction. They may conceive of potentiality, or matter as contradictory.

This sounds incredibly wacky. For one, even if there is some metaphysical violation of Excluded Middle, that doesn't preclude it from human understanding nor does that make reality "violate the laws of logic" because there are not "the" laws of logic. There are many such sets of laws, and some drop Excluded Middle. It would certainly be a surprise to the Intuitionists that they don't understand constructive mathematics or their own logic because it doesn't assume EM as an axiom. — MindForged

it's not "incredibly whacky" it's the central point of our discussion. There are always things which escape human understanding. They are things which human beings do not understand. I would place "infinite" in that category, but you seem to think that "infinite" is understood, so let's go back four or five hundred years and say that at that time, "infinite" was not understood. The reason why it could not be understood was that it appeared to defy the law of non-contradiction, in the form of paradoxes. When some premises lead to conclusions which contradict what we know as basic fact, then there is a problem in our understanding. These are things which escape human understanding.

When this problem occurs, there are two principal possibilities of the cause of the problem. One is that the fundamental premises, how the terms are defined, are faulty, the other is that the fault is within the logical process. So our subject is "infinite", a few hundred years ago,when it escaped human understanding. The term produced problems which caused the appearance that it could not be understood. The logicians at the time decided that the best way to proceed was to change the premises, the defining terms of "infinite". What I am arguing is that misunderstanding is not due to faulty premises, but to faulty logical process. Zeno's paradoxes deceive the logician through means such as ambiguity or equivocation, by failing to properly differentiate between whether the aspects of reality referred to by the words, have actual, or potential existence. That's what Aristotle argued. So the logician gets confused by a conflation of actual problems and potential problems, which require different types of logic to resolve, and are resolved in different ways. Instead of disentangling the potential from the actual, the logicians took the easy route, which was to redefine the premises. All this does is to bury the problem deeper in a mass of confusion.

And this is all besides the point anyway. The nonsense you tried to pass off earlier was the idea that there are "defining features" of things like parallel lines despite now knowing that these terms are defined by the user's (implicitly or explicitly). They don't have inherent definitions, they're defined within a certain domain. So the idea that you tried to push that parallel lines don't intersect is purely based off the underlying assumptions of the geometry in which you made an assumption of, it is not true writ large in geometry. The Parallel Postulate is only true inasmuch as it's assumed to be so in a geometry and anyone saying otherwise is just misinformed about how mathematical formalisms work. — MindForged

You haven't addressed the issue here. You only support these claims with a reified "space", assuming that space is a physical object to be studied, and not a conceptual object. It's beginning to appear like this is the crux of the differences which we have. Do you distinguish between objects which exist solely in the mind, imaginary things, concepts and ideas, and things which have physical existence in the world, like rocks and trees, and planets? If so, then when we use words to refer to things, we ought not confuse whether the thing referred to is ideal, a concept, or a physical object. An infinity, just like infinite, is always an ideal, a concept, never something in the physical world.

I haven't reified anything, I didn't treat these as anything other than abstract mathematical constructs. — MindForged

What's this then?

Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). — MindForged

See, you are treating "space" as if it is something described by geometry. In reality, since we can use various different geometries to describe the various types of objects we sense, there is no such thing as "space". We might be able produce a concept of "space" from this geometry, and another concept of "space" from this other geometry, but it really makes no sense to talk about "how space is", or "if space is curved...", because there is no such thing as "space", not even as a concept.

This is why your geometrical examples are irrelevant, and way off the mark. You are talking about geometry as if it is created to describe some sort of "space". Then you need to bring in some principles to account for the intuition that space is in some sense infinite. However, this is totally uncalled for. We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects. It is only when we approach these concepts of geometry attempting to synthesize a concept of "space", that we find "infinite" as inherent within the principles themselves. Then there is a problem because we want to produce a concept of space which allows for the infinite because we have been confronted with the infinite as inherent within the concepts used to describe objects.. But this is only a pseudo problem. The infinite is not part of the objects which we measure and therefore it ought not be part of any concept of space. The infinite is only a part of the mathematical tools which we use to measure with. So any attempt to bring the infinite into our conception of "space" as a the thing being measured is a mistake.

Infinitesimal points were introduced to allow that a multitude of infinitesimals may have dimensionality, therefore a continuous line could be conceived of as being composed of infinitesimal points. — Metaphysician Undercover

"Infinitesimal point" is self-contradictory--points are, by definition, dimensionless and indivisible; infinitesimals are, by definition, dimensional and potentially divisible without limit. As I have stated repeatedly, in my view an infinitesimal is not a discrete unit of any kind, and a continuum is not composed of infintesimals.

Isn't that sufficient for a continuum model? — TheMadFool

Yes, as I have acknowledged, the real numbers serve as a sufficient model of a continuum for almost all purposes within mathematics; but it is still a model, not a true continuum itself.

Infinitesimal point" is self-contradictory--points are, by definition, dimensionless and indivisible; infinitesimals are, by definition, dimensional and potentially divisible without limit. As I have stated repeatedly, in my view an infinitesimal is not a discrete unit of any kind, and a continuum is not composed of infintesimals. — aletheist

That's exactly the problem with infinitesimals, their dimensionality is ambiguous. If it is not a discrete unit in any way, then it has no form and therefore no specific dimensionality. However, you say that it is not dimensionless. Therefore it has dimension but its dimensionality is completely undefined and ambiguous. But dimensionality is purely conceptual, and must be defined. Now we have an infinitesimal which is defined as having no specific dimensions yet it is not non-dimensional. Therefore it is either completely irrelevant to our conceptualization of dimensionality, or it occupies some vague, ambiguous, undefined position within our conceptualization of dimensionality which could only serve to mislead us.