The three laws of thought, as put forth by Aristotle, are analytic propositions, not because of their content, but because they are true necessarily, which just means their negation is impossible — Mww

Is this something you believe? Because intuitionistic logic alone has existed for nearly a century now so it feels like a strange view to me. Logical operations such as negation are defined by a logical formalism, they don't stand independently of them, and so how they function is not something we can say works a particular way outside of a logical system. In intuitionistic logic, Excluded Middle is not necessarily true because negation is defined differently and so Excluded Middle cannot be asserted within a universal quantifier. It's not that the negation of Excluded Middle is true, but that EM is not a true in all models.

I didn't at all comment in Galileo. What I called stupid was your insistence in childishly declaring the most held philosophical position on the matter as being "objectively" false as if it were just an obvious and incontestable thing to say. You are proving yourself not to be a serious person.

I'm not knowledgeable on Nietzsche but based on what you said it sounds in part correct but it ends at the wrong conclusion. It's true that what we essentially do is create models of the world and that there's some simplification. But that doesn't make the enterprise false, it just means we need to speak more carefully. There's a reason statements like "All models are false but some are useful" exists. Based on how well they perform on various theoretical virtues we think it more likely that we are getting closer to the truth.

I thought that was the definition of a synthetic proposition though? A synthetic proposition is true by virtue of the meaning of its terms and due to facts about the world, whereas analytic propositions are true by virtue of the meaning of their terms alone? — philosophy

Nah, just check the SEP:

An “analytic” sentence, such as “Ophthalmologists are doctors,” has historically been characterized as one whose truth depends upon the meanings of its constituent terms (and how they’re combined) alone, as opposed to a more usual “synthetic” sentence, such as “Ophthalmologists are rich,” whose truth depends also upon the facts about the world that the sentence represents, e.g., that ophthalmologists are rich.

A synthetic proposition is one which is true by virtue of the meaning of its terms and facts about the world — philosophy

I think you typoed. Synthetic propositions are *not* true in virtue of meaning.

In any case, calling these the "laws of thought" is and has always been kind of dumb. Not only do they not correspond to the limits of thought - people have inconsistencies in their beliefs all the time, and occasionally don't seem to hold to Excluded Middle - but they're just one possible set of axioms a logic can adopt. There are logics without each of these axioms.

That aside, this idea is full of holes. I'd have to go check what exactly it was, but even this "in virtue of" relationship has faced technical roadblocks in the past.

Moreover, these axioms are not true in virtue of some kind of indisputable way. Consider the Law of Non-contradiction: a proposition cannot be true and untrue or equivalently the negation of a conjunction of a proposition and it's negation is always true. It's clear this is an exclusivity condition for propositions and the truth-value they relate to. But mathematically we know that this type of setup is not all that exists. We can go beyond functions and use broader relations. If we take truth-valuation to be a relation between proposition and any number of truth-values then it's clear that Non-contradiction no longer holds necessarily since the technical machinery exists in that logic to violate it coherently. Some proposition 'P' could relate to truth and relate to falsity, and so it's clear this logic lacks that exclusivity condition since the semantics of truth value assignment has changed. Of course, you'd need to find something that can fill the overlap which is pretty controversial.

So clearly there's more going on here than meaning establishing truth. The terms employed in the definition give away the assumptions that seem to guarantee the truth, but those assumptions could be attacked it questioned anyway. There's broader issue with the analytic-synthetic distinction but since others are likely to mention them I went a different route.

This never seemed like a real argument to me and none of those solutions seemed quite right to me. The obvious answer always seemed to me to be that the whole premise is faulty. If God is defined as omnipotent, and omnipotence is understood as the ability to do anything, we should recognize that there's an implicit assumption there: This only regards things that can be done. Why would Christians define God as a being who can do things that can't be done in principle? In this case, that impossible thing is having a being that can succeed at any possible thing failing to create a scenario where he fails to do something. That screams contradiction to me.

There might be an interesting discussion with regards to God failing to do things we are capable of but at the very least the rock thing always feels a bit silly to me. It'd be like saying "If God can't checkmate from both sides of the board in the same game he's not omnipotent."

Sadly one cannot appeal to mathematics without begging the question — sime

Unless you can actually provide an alternative mathematical formalism that is as credible and useful as standard classical mathematics this is a nonsense claim. Alternatives exist but with rare exception they are of philosophical interest as opposed to mathematical ones. It's not begging the question to use the most accepted and applicable theory in the relevant details. It's quite literally the opposite.

But the only way of assigning time intervals to each term of the associated geometric sequence, say {0.5,0.25,0.125...} is to ignore the application of zeno's argument to each and every term. — sime

How does it ignore it? Doing calculations with infinities (even the continuum) is old hat now, real analysis and calculus exist. You can argue whether or not this model is the best model for real space, but currently all dominant and used models of spacetime treat it as a continuum and is understood by classical mathematics just as the rest of science has it's underlying math understood via ZFC.

And the only way to reach the value of the bound is to literally sum an infinite number of terms, which assumes the existence of a hyper-task that mathematicians do not possess -unless , say, motion is considered to represent such a hyper-task, which is precisely what Zeno's argument calls into question. — sime

Again, if you just object (sans argument) to the most useful mathematics in history that's on you. But summing an infinite number of terms can be done in calculus. It isn't even clear that the base assumptions that go into Zeno's paradoxes are true. You haven't argued for them, and most of the time their justification is just treated as somehow obvious despite the conclusion being demonstrably false.

I mean let's just make it fun. If a hypertask were impossible, there would be no motion or change. There is motion and change. Therefore hypertasks are not impossible. And given our best models of spacetime characterize space consistently with this conclusion, it's not like you're presenting a serious empirical issue with infinitely divisible space. You either have to point out a logical impossibility with infinity (but no contradictions exist), explain why infinity is perfectly acceptable in math but not in models of the physical world or you need to show what observations ought to make us change our models of spacetime as a continuum. Otherwise I don't see how your position is supposed to hold any merit over the standard one.

Ehh, that's not a correct representation of what's wrong with Zeno's paradoxes. The fact that an actual arrow may be fired (or one may outrin a tortoise) shows that something is wrong but it doesn't tell you what is wrong. Aristotle claimed the problem was the idea that an actual infinity (the division of space infinitely) was possible was the false assumption in the argument. However, we know believe Aristotle to have been very wrong. Distance can be infinitely divided, but the time it takes to cross that each segment decreases appropriately so it sort of cancels out.

This wasn't understood until we had calculus and developed a formal understanding of infinity via Cantor, Dedekind, Balzano and co., so it's definitely not a trivial discovery.

You mean philosophers like Popper, who wrote literally books refuting empiricism? - — Inis

It's strange, it's as if you think philosophy is done by a few major names whilst ignoring progress in the field in the more than 50 years since Popper (and far longer for Kant) were anything like representative of the current state of the debate. This is absurd.

GRW is not quantum mechanics, Bohm has been refuted so many times it's getting boring, Copenhagen is psychology. These are all standard views in foundations of QM. — Inis

Bohm hasn't been refuted, Copenhagen isn't psychology (no matter what you think about wave function collapse you're misrepresenting it) and neither is, in terms of empirical support, any better than the other currently. But all of that is besides the point (which you ignored) that no one short of an outright ideologue talks like this innthe actual debate about these views.

I hope you appreciate the irony in your appeal to thought experiments to defend empiricism! — Inis

I didn't mention or use any thought experiments at all. A prediction made by a model is not a thought experiment. The models say X will be seen under Y conditions, but we don't see those in the cases where we apply Relativity and QM models outside the domain they've been successful.

You are confusing the two theories, for which there is zero empirical evidence against, with theoretical problems encountered in the attempts to unify them. Empirical evidence is literally irrelevant at this point, it's all about theoretical consistency. Ask a String theorist. — Inis

Can you keep track of your own points? You mentioned the inconsistencies between QM and Relativity and claimed there was no empirical evidence against either but that's not true. The empirical evidence against them isn't even up for debate in the domains they weren't made for (Relativity for quantum scale events and QM for macroscopic events). This point had nothing to do with QM interpretations, that was the previous point regarding your ridiculous way of speaking and dismissing other theories (or interpretations) in a way no professional would. That's on the level of ideological attachment (or rejection, in this case).

Given that Empiricism, the doctrine that knowledge is derived from the senses, is objectively false, I would hope we could get beyond it, if not transcend it. — Inis

This kind of statement sounds really unthinking. Making such grandiose statements as if they were trivialities is not a very good way to argue. If empiricism were objecrively false one would expect this to be represented amongst philosophers views. But the opposite is true, according to the PhilPapers survey more philosophers are empiricists than rationalists by a decent margin(8-9%), and those who are neither strictly empiricist nor rationalists out number both the rationalists and the empiricists. If this does this at all sway you to speak less assuredly I'd be rather surprised. No one would talk about views in, say, quantum mechanical interpretations this way so doing so here frankly sounds stupid.

Inconsistencies are the greatest flaw in any theory, rendering them immediately problematic. Famously, right now, we have inconsistencies between theories, rendering each problematic, despite there being zero empirical evidence that either theory has problems, and no one can even come up with a suitable thought experiment. — Inis

This doesn't seem quite true. Several inconsistencies between the theories results in false predictions when applied in each other's domains, yes? Applying general relativity at quantum scales results in infinities we can't renormalize and applying quantum mechanics as cosmological scales predicts fields with energy levels that would result in enormous black holes, and neither of these are observed.

No one names the leaves of a shamrock as if they are different entities. It would be like Clark Kent really believing he was a different being a Superman and then not understanding why people are confused by that idea.

it's not a concept, it's a number. Remember how I said the size of the set {1,2,3} was the cardinal number 3? The size of the natural numbers is the infinite cardinal number Aleph-null. It's not a concept, it's a number.

The reason the alephs can add any finite number without increasing size is because if how size is defined. As I said, a set A with the members {1,2,3} has a cardinality of 3. Let's take a larger set, an finite set. Take N, the set of Natural Numbers. {0,1,2,3...}. Then let's take E, the set if Even Numbers. Which set is larger, N or E? Well we can do this by trying to make a bijection between the sets, that is, by mapping each element in one set with exactly one element in the other set. In finite sets, it's obvious when a set is larger or smaller because one set will run out of members to map together. Like take set A from above and compare it to set B with the elements {0,1,2} (set A on the left, set B in the right):

0 - 0
1 - 1
2 - 2
3 - (no more elements to map together)

So we know set A is larger, it has more members. But what happens when you try this with the Natural Numbers and the Even Numbers? Intuitively it seems like the natural numbers should be twice as big since the Even Numbers are missing half the numbers (the Odds) which exist in the set of Naturals. But that's not what happens:

0 - 0
1 - 2
2 - 4
3 - 6
4 - 8
Etc.

No matter how far you get into the set of Naturals, you'll always have an Even number to match up a number from set N. This means the Naturals are the same size as the Evens despite lacking the Odd numbers. And that's why adding 1 (or any other finite number) to an Aleph won't change the cardinality. You'll still be able to map elements from the original set to the set + 1 meaning they are the same size. It's still a number, but the way size works means that if the elements of one set can be completely mapped together with another, they are the same size even if your intuition tells you it shouldn't be that way. The logic doesn't show any contradictions arising.

The size of sets are numbers called Cardinal numbers. Like some set A with the members {1,2,3} has a size of 3. The size of the set of natural numbers (whole numbers 0 and greater) has a size of Aleph-null, the first infinite number. If it's a set, it has a size.

The infinity symbol isn't a number, but there are infinite numbers. My point was that what one is doing with limits (where you see the infinity symbol) isn't actually about the set theoretic understanding of infinity, it's just an unbounded sequence as opposed to a definite value. But the cardinals and ordinals are definite values and infinite.

So an axiom of infinity is effectively 'when you change it, it does not change'. What sort of reasonable system of the world would adopt such an axiom? Where is the evidence for these magic objects that can be changed and remain unchanged? — Devans99

"∞" isn't a number. Aleph-null is an infinite number. And again, infinity does change. You just cannot add or remove *finite* amounts of it to change it because it's the definition of finite that it changes by finite modifications to such a value. If you take the Power Set of Aleph-null, it increases and becomes Aleph-One, the size of the continuum, a larger infinity. But that's because I'm adding by infinite amounts, that's what let's it change. And it gets different when you get to the Ordinals too. Adding finite to the infinite Ordinals does change them.

As for what I assume you're asking for (real world examples) we can take space or time. As far as current models go, there's no fundamental unit of space or time, they are continuums. So if I have some slice of space I can always zoom in by some arbitrary amount (even an infinite amount) and there will always be more space or time.

I could have pulled from all kinds of other sites, just grabbed that one. — Rank Amateur

None of which will tell you that limits make use of infinite numbers, because they don't. The "infinity" mentioned in limits just means "some arbitrary number greater than any yet reached in the sequence". The point of the limit is to avoid infinity, really.

Tranfinite numbers are not, by definition infinity, They are, by definition < infinity, one thing can not, be less than something and be the same thing as that which it is less than. — Rank Amateur

There is no number called infinity. Infinity is a type of number. Transfinite numbers are infinite numbers. The term 'transfinite' is just an old term, you can call them the infinite Cardinals. They are by definition infinite, they can be placed into a bijection with a proper subset of themselves. An infinite set with one less member is not the same set. It lacks the member removed from it. What doesn't change is the cardinality (size) of the set.

That's talking about limits at infinity in in calculus, not the actual infinite numbers. Limits in calculus are usually defined as something like,

L is the limit of f(x) as 'x' approaches infinity if f(x) becomes arbitrarily close to L whenever 'x' is sufficiently large.

I could just as easily replace the term "Infinity" here with whatever else I need. The point of the limit is that is grows arbitrarily large or small, it's not a definite number. Transfinite numbers are outside the domain and range of real numbers used in calculus, so it's just not the same thing as the infinity under discussion. All real numbers (as in decimal numbers) are finite.

However, the transfinite cardinals and Ordinals are numbers and are universally acknowledged as infinite numbers. They are larger than any finite number and are not limits. They are sizes and order numbers of infinite sets.

It clearly does give the wrong results. There are more numbers than squares in any finite interval. So we can induce this applies to all intervals. — Devans99

Explain how this follows. You're using induction to generalize in a way that seems ridiculous. We can always find new squares to map on to so I don't know where you're getting this idea that "it clearly does give the wrong results".

Although they're equivalent, I've always rather liked Dedekind's description of infinity. I think it's a lot easier to (for want of a better word) picture infinity that way (as a bijection between a set and a proper subset of itself). Probably because it's easier to show it, it's how my professors often spoke about it so maybe that just stuck with me

Wrt physics, there's some evidence that attempts to make discrete models of spacetime might not be feasible:

https://www.nature.com/articles/nature08574

At the very least, the prospect of giving up Lorentz Invariance seems difficult given this. Not a death knell, loop quantum gravity (speculative though it is) is hardly refuted currently. But the standard model treats spacetime as a continuum so that seems to be the best assumption for now.

You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time. — Devans99

Cool. It's a a good thing "A number larger than any number" is not the definition of infinity in mathematics. The closest correct description of infinity that resembles what you're saying would be to say every infinite number is larger than any finite number. There's no logical error that results from infinity.

To be up front, my undergrad requirements for physics didn't really get into QM, so my "knowledge" of QM is a hackjob accrued from friends, colleagues and the Google box linking papers.

All that said, I'll give it a go. From the name, many people think QM must say space and time are quantized (discrete/, but many values in QM are continuous (position for instance), and space and time are among those values. In and of itself that doesn't mean too much, since you probably could modify the theory to use discrete values for these instead (although in practice there's no benefit to doing so).

If there's a minimum distance you start messing up a lot of other things in physics, especially as it relates to Relativity currently, such as people in different reference frames measuring different Planck length due to relativistic effects. S you'd probably have to drop Lorentz Invariance, but somewhat recent experimental observations (2011) haven't borne out high enough violations of it that would be expected if spacetime were discrete at some scale:

https://www.nature.com/articles/nature08574

Planck length and time are just measurement limitations at best. It's plausible that Planck lengths are the smallest measurable lengths, but there's no current reason to thinks it's a fundamental chunk of reality. Like these are somewhat taken arbitrarily. Inagine I was measuring things based on, I don't know, the radius of the Earth. That wouldn't mean everything else should be measured as if they were a multiple or divisor of the planet's radius. Natural units are all well and good. Some make a lot of signifance out of the Planck length but all we can really say that isn't speculation is that it sets a limit on the non-negligible effects of Quantum gravity. Our models of quantum gravity would only work down to that scale, so we'd need something more to model anything that exists at a smaller scale.

OK so thats equivalent to saying 'I have this set to which I can add to and the size does not change'. Thats nonsense - anything you add (non-zero) to the size changes. That should be an indisputable axiom of mathematics or at least derivable from simpler axioms. — Devans99

And? That's not a contradiction. Size is understood by the theory of cardinalities, not the intuitive idea you're working from. It's not a contradiction at all. Anything FINITE that you add to has it's size change. In fact, that's the very definition of something which is finite: it changes size when additions or subtractions are made to it. And so too with infinity, it's very definition entails it does not change when some finite amount is added to it.

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. — Devans99

It does change. The set has an element it did not have before. But the cardinality does not change. Derive the contradiction or just admit that you cannot. This is not a logic error, you're just using a dumb definition of infinity and wondering why people aren't using it. It should be obvious why mathematics does not use your definition, while you pretend to have found an area where maths is 'illogical'.

But we have good evidence that the speed of light is constant and the rest follows. We have no evidence for 'stuff that we change that does not change' and it makes no logical sense anyway. — Devans99

That's false though. We have both mathematical evidence and current scientific evidence suggests that space and time can be arbitrarily subdivided without reaching a discrete unit.

Infinity is not a number of any kind it is a label for a concept. — Rank Amateur

It's a number and a concept. Of note is that the concept is best understood by working out the properties of the numbers. Namely, the transfinite Cardinals and Ordinals, which are infinite numbers.

Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? — aletheist

I should clear this up. I posit that those models, well evidenced as they are, are the best explanation of why we make the kinds of observations we make (e.g. not reaching any sort of discrete unit of space no matter the magnification). And since both QM and Relativity have some type of continuity to spacetime, we ought to accept this until such time as we have reason not to. I don't mean I completely see the infinite totality if a thing, but that whatever credence we give to observations in establishingnor refuting the actuality of infinities, our current observations of space and time don't seem to contradict this possibility at all.

there are no infinite numbers. There is no greatest number (because X+1>X), so there can be no number larger than any finite number. — Devans99

Infinity is not defined as the largest number. Stop saying that. That is not the mathematical definition of infinity. You keep repeating yourself and ignoring the corrections. If you repeat some crap about infinity being defined as "the largest number" I'm done. Just asserting there are no infinite numbers is a stupid argument.

I am not using colloquial definitions; I'm doing my best to be logical about it (unlike Cantor).

You literally keep repeating that infinity is defined as "the largest number" when no one else here has said so and it's not the mathematical definition of it. Cantor was a celebrated mathematicians (eventually) and unlike you showed his actual proofs and gave rigorous definitions and worked out the consequences to show no contradictions arose. You're doing the mathematical equivalent of shitposting.

But infinity cant't be bigger than any number because then it would not be a number. That's the mother of all contradictions. — Devans99

It's not bigger than any number. Stop stop stop using colloquial definitions when talking about a formal discipline. Infinite numbers are larger than any finite number, there is no infinite number larger than all infinite numbers. You don't know what you're talking about.

Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion. — Metaphysician Undercover

I didn't assert a contradiction. Your claim was that you have to be able to count the series in order to declare an end point, which is false. A race can be run backwards, it can be run from the middle out to either end, etc. The order is irrelevant. A race doesn't end at a random point, the end is defined when the beginning is defined.

Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question. — Metaphysician Undercover

Disingenuous. The point is you cannot define it as necessarily impossible and then claim to have proven it to be the case. Induction only yields probable conclusions, you claimed the conclusion that infinity was impossible to actualiz was necessarily false, and you brought up inductive generalizations to prove that. You made a non sequitur, induction cannot give you necessary conclusions.

So, what kind of infinite thing (infinity) do you think you could observe? — Metaphysician Undercover

Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible.

The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter. — Metaphysician Undercover

You say things like I can't just quote what was said before. The question was whether or not geometry was about measuring things. I said that was the canonical use of the discipline, but that's not what the discipline itself was about. That you're trying to claim the division being pointed out avoiding the issue is ridiculous. I brought it up because you said this:

We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects — Metaphysician Undercover

But that's absurd. We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. The common assumption was that reality could not be any other way than as a model of a Euclidean Space until Non-Euckidean geometry came along and Relativity gave credence to thinking our actual space was best modelled as a pseudo Riemannian space. Funnily, Euclid made as a base assumption in his geometry that space was an infinite plane but I'm sure you'll object to that without question begging and ignoring that Euclid's assumption contradicts your claim that geometrical axioms are about measurement. It's about studying abstract math structures of a certain kind.

Anyway, theory and application aren't the same and the idea that geometry is fundamentally about measurement is wrong. If you think otherwise, show where measurement appears in the formalism of common geometries.

Why are you quoting Oxford about a mathematics concept? Aside from the fact that the Oxford definition isn't really contradictory, the mathematical definition of infinity has no contradictions. A set has an infinite cardinality if the set can be placed into a one to one correspondence with a proper subset of itself. No contradictions at all. Cardinal numbers are numbers, so infinite Cardinals are numbers as well.

There are two defined start points, 0 and 1, with an infinity of points between them. No end point. — Metaphysician Undercover

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. The only way your argument could work would be by the hilarious assumption that the quantity of real numbers between any arbitrary interval were finite. Defining the start and end of something does not mean that end is not an end point. For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over.

guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite. — Metaphysician Undercover

And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning.

But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence. — Metaphysician Undercover

The problem is that is not the exclusive colloquial definition of infinite for reasons I've already mentioned.

If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? — Metaphysician Undercover

What you seem to be missing is that the math is used to model the world and so far no model of finite space or time has any particular empirical or theoretical backing. You'd have to pin your hopes on something like Loop Quantum Gravity, but that's highly speculative and has about as going for it currently as String Theory does (not to say it won't change), whereas time and space are still modelled as continuums in both quantum mechanics and relativity. If the current models are accepted, it's just a performative contradiction to give them credence but to arbitrarily say some of their fundamental assumptions are to be presumptively excluded from reality.

Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility — Metaphysician Undercover

This is just false. Mathematics already resolved Zeno's paradoxes so clearly adopting the mathematical models of spacetime does not create paradoxes given we know how to resolve the apparent issues Zeno saw with having them be infinitely divisible. Zeno made fundamentally mistaken assumptions about the consequences of trying to cross an infinite series, they turned out to be negated.

What you're not respecting, is that for Aristotle ideas are part of reality. — Metaphysician Undercover

Word game. By "reality" I meant being actual. You've already said you don't think this is possiblefor infinity, I was showing you how you were holding a hypocritical standard that only applies when other people use definitions you don't like, but you're perfectly find doing it yourself even if it's not the colloquial, "true" definition.

Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible? — Metaphysician Undercover

Contingency and possibility are not the same thing. Necessary truths are also possible truths (because possibility just means truth in at least one possible world). Contingency means some modal proposition is true in some worlds and false in others. That aside, I don't see the relevancy in your questions about modality. In nearly all cases, potential is just a synonym for the term "possible". E.g. Every English speaker can readily understand "I have the potential to be a doctor", but sentences like "I have the potential to be a doctor but I cannot be a doctor" have to be disambiguated since it switches between two different types of modality (logical possibility and physical possibility), otherwise it's a flat contradiction. You can consistently say "I have the potential to be a doctor but in actuality I functionally cannot".

If you say X is a potential infinity but it cannot be actualized you are either contradicting yourself or you're switching between 2 different modalities. If it's the former, well that's not workable on pain of absurdity. If X cannot be actualized it's not a potential anything, the label doesn't fit. If it's the latter, then you're playing a shell game. You have to argue that Infinity isn't metaphysically possible, it's not contradictory so it's not inherently off the table for a consistency issue.

You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that? — Metaphysician Undercover

Dude, just prior to the part you quoted I said that the canonical application of geometry was for measurement:

, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you). — MindForged

My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing.

Once an AI has the freedom to evolve and improve itself, there is no predicting what it might do. — Pattern-chaser

We already have self updating programs now and the world hasn't ended. This sort of ability to self refer and self alter isn't a problem in and of itself, it's where you place them, what you have them do and what sort of checks there are on them. There were a number of times automated systems nearly caused either the U.S. or the USSR to deploy nukes in what they thought was retaliation of an attack the other side initiated. Luckily human oversight stopped that.

Like the whole idea of SkyNet (or whatever movie has nukes controlled by an A.I., I know Terminator isn't the only one) is really stupid. Keep the A.I. on an isolated network that isn't connected to an outside network nor responsibile for anything that can cause too much trouble (e.g. no access to factories where it can place orders on what is built if you're making a general purpose A.I.) and there's really not much to worry about other than a lot of goofs as the system alters it's state. Like just watch videos of neural nets learning to walk. While it's cool that they can, eventually, do it, it's so obviously not very good in comparison to the real thing right now.

Maybe there are ethical concerns at a certain point but I see 99% of "worries" about A.I. to be overblown or else have very obvious precautionary measures that can be taken.

Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED. — Devans99

You do not understand the concept of cardinality, do you? The size of the interval is the size of the continuum, aleph-1.

Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity). — Devans99

Why do you keep asserting this as fact? That is not how "infinity" is defined in mathematics because it's too hazy and informal a definition. An infinite set is, e.g. the transfinite Cardinals, a set whose members can be put into a one-on-one correspondence with a proper subset of themselves. There is absolutely no mention of being "greater than any assignable quantity, you're just wrong. Find one mathematics textbook that formally defines and describes infinity that way. Go on, I'm sure you can do it... (Obviously I'm being sarcastic here).

When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua. — Devans99

That's not an argument, that's a statement that you cannot justify. As it happens, it's perfectly understandable why the CARDINALITY doesn't change. The set does change if you add a new unique element, but the size cannot change by mere finite additions because we can still world new numbers to put into a function with the new element that was added.

I was just helping a friend out with a website. The RGB color values could be changed by the file in question as needed, but there seemed to be an oddity. Because color values are cappped at 255 per channel, adding any number to that channel resulted in no change to the value of that channel. I suppose 255 must not be a quantity then since adding to it did not cause the value to change. Saturation mathematics must be incoherent despite it's use in computer science then!

It's a little funny that mathematics by unjustified dogmatic assertion went out of vogue for everyone besides the ultrafinitists.

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". — Metaphysician Undercover

This is mistaken. My point is simple. Infinity is often intuitively understood as unbounded quantitatively. In other words, given any arbitrary number N there is some number N+1 that can be accessed from N. No set end point, basically. However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. So clearly the colloquial understanding of this infinite is not consistent.

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. — Metaphysician Undercover

I didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? All that tells us is that those observations are finite and so the next ones made will likely be finite. It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible. We have the math to make perfectly logical sense of this and all current physics assumes this is the case (even if you wanted to suggest loop quantum gravity, I could suggest the equally speculative string theory where space can be infinitely divided again).

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. — Metaphysician Undercover

I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object.

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, — Metaphysician Undercover

Whatever Aristotle may have said, refer to what you said before:

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.

You said it's impossible for humans to understand and yet clearly you think you can explain what about it makes it impossible for humans to understand. So unless you can explain something about things you can't possibly understand it sounds like you're contradicting yourself.

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. — Metaphysician Undercover

You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have.

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. — Metaphysician Undercover

I never said space was physical. An object, sure. It has properties after all and we have studied these properties. I'll come to this in a moment.

What's this then? — Metaphysician Undercover

, 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. — Metaphysician Undercover

I don't really see how you are saying anything here because you're moving between unrelated points. When I referred to "how space is" I was talking about the actual structure of spacetime, not the more vague, general concept of space. Some abstract geometric spaces are curved and some are not. Whether or not the structure of the actual space of the universe falls into one or the other is a physics question, and current physicsand observational evidence says space is curved. If you cannot even accept this there's no point in this.

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" — Metaphysician Undercover

Er, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you).

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 — Metaphysician Undercover

I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me?

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".

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.

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).

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.

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.

"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.

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.