You could be right. And he's still waiting for answers.

Actually, I was too rash in stating that Flannel Jesus' reply did not call on any infinitesimal values, ie, did not depend upon h. Of course, his conclusion does depend upon defining the infinitesimal as h, and thus falls within the scope of the hyperreal number line, within which context 0.999...=1, as I asserted in my original post.

I'm sorry, but none of the replies so far seem to evidence any familiarity with number theory or basic set theory... in fact, I can't really identify any rational position to which I can respond... is there anybody out there who is familiar with this area of mathematical philosophy? I mean, when a respondent asserts (without supporting arguments) that number lines don't exist, how does one begin to frame a reply?

Number lines do not exist (or, at least, cannot be described).

Wow, I did not see that coming. More than 2500 years of mathematical development flushed down the toilet in a few seconds!

The real numbers as intrinsic values are a phantom; an illusion.

What are your supporting arguments? Give me some help here, I'm trying to understand your position.

All definitions are of the form: X is not(Everything Else)

The assertion that X has no identifying properties in its own right is certainly a courageous approach. To my knowledge, nobody in the previous history of mathematics, from Euclid to Penrose, has ever adopted such a definition. By this method, how would you define "prime number", for example?

When we describe numbers we actually describe the relationships of numbers.

"Relationships of numbers" is a defining property of the relational number line (the line of negative and positive integers). But you deny the existence of number lines. Can you develop this point?

The example provided by Flannel Jesus conforms with the parameters of the hyperreal number line, as set out in my original post. Of course, it does not call on any infinite or infinitesimal values, but if he wishes to argue therefrom that it is consistent with the logical properties of the real number line, he will need to provide clarifying arguments.

TC: I'm sorry, but "We'll have to trust Chat GPT" is not a philosophical argument. I'm not looking for someone to tell me the "right" answer; I am posing a question in the (apparently vain) hope that someone out there actually understands mathematical philosophy.

Forgive me folks, but the title of the post reminds me of an old, old joke: a Polish person goes for an eye test. The optometrist says, can you read this chart? And the patient says, "Read it? I know the man!"

Actually, nude bathing is my favourite form of naturalism... (shut up, alan)

"A mathematical point is a definition of nothing. We can't use 'nothing' as a building block."

The second sentence is granted, but the first sentence is not immediately intelligible. Suggest review Robinson's "h" and reconsider.

native-born Irish American — BillMcEnaney

Ain't no such thing. Choose one or the other.

"I believe a fixed percentage of men are born gay. It's not a large cohort--maybe 2% or 3%."

I think 5% is likely a more accurate figure, although there is obviously some leeway, depending on the criteria.

Genetic science and anthropology can explain why some dispositions, though apparently dysfunctional, do not disappear but settle down to around 5%. This applies across a wide range of phenomena from alcoholism in laboratory rats to colour vision deficiency in humans.

This raises the question whether a 5% deviancy rate in any phenotypic feature has a survival value for the species.

In the case of colour vision deficiency, this is a topic of ongoing research. Experimental evidence suggests that CVD people have a heightened ability to distinguish fine shades of green and brown, and heightened night vision. There is anecdotal evidence that in Vietnam, CVD soldiers were sought after to act as foward scouts, because of their heightened ability to "see through" enemy camouflage.

One may also speculate that in hunter-gatherer societies, homosexual people, who have the same nurturing and loving instincts as heterosexuals, may have been valued as carers for the children of parents taken away early by disease or predation.

I recollect that Dawkins tells us somewhere (was it in The Selfish Gene?) that according to computer modelling, the survival of the species is best guaranteed if 70% of females are faithful to one partner, and 30% of males are faithful to one partner; interpret that as you will.

Another provocative idea (not due to Dawkins) is that the Oedipus and Electra complexes evolved as another survival mechanism, whereby the parents would teach their children how to reproduce before they go out into the world to start their own families. But try selling that idea at your next seminar.

"But, for the moment, cannot tell us anything useful about how dark matter prevents the collapse of stares near a black hole,"

Apologies for the bad grammar. Must proofread more carefully.

I would say that, at the current level of our technlogical development, ChatGPT has got it just about right... but that in ten or twenty years more, perhaps not.

You don't need AI to calculate large sums of data. We've been doing that since the Romans invented the abacus. (Little-known fact: expert abacus operators in Hong Kong could outperform the original IBM PC-XT!) .

AI comes more to the foreground in running thought experiments, but at this stage it does not seem to outperform human effort. Thought experiments, by definition, involve conceptual scenarios which go beyond any prior experience, and AI is (for the moment) not very good at this, except in certain very specialised fields like anti-biotic research, where its development has been very intensive. But, for the moment, cannot tell us anything useful about how dark matter prevents the collapse of stares near a black hole,

For the moment, the main deficiency in AI (where philosophy is concerned) is its inability to formulate and argue a strong, original case. Presented with a philosophical question, its responses too often resemble a summay of discussion points.

I don't know what replies the question may have garnered in other forums, but I was hoping for something a bit more interesting than my own in this one... my Dictionary of Philosophy doesn't list "snippy" (must be out-of-date)... the question is actually deeply interesting, and cuts across many lines of philosophic thought, from Euclid to J L Austin. Surely somebody out there has something positive to contribute?

A further (whimsical) thought: if we understand "terms" to include symbols, what is the etymology of "+" and "x"? Have you noticed that "+" refers to a singular addition (as in, 2+3=5) but, by turning "+" through 45 degrees, it transforms to an iterative addition (as in 2x3=2+2+2=6)?

A valid question although, to be honest Tony, unnecessarily verbose!

The question you raise is central to Fermi's Paradox. it could be condensed as follows: suppose there is just one other civilisation comparable to our own, and it happens to be on the other side of the universe, billions of light years away. This would resolve Fermi's Paradox, since we would NOT be alone in the universe; but the other civilisation would be so far away we would have no means to communicate with it or even discover its existence, unless we can find out some alternative physics, such as diving into a wormhole, which at this stage is pure fantasy. Light and information from such a civilisation would take at least 13.75 billion years to reach us; to reply, or travel there (at the speed of light!), would take the same amount of time; just do the sums.

I vote for "axiom" as the term which, as much as any other, crystallises this question. For the ancient Greeks, "axiom" meant a proposition for which nobody could think of a proof, but the truth of which had to be accepted on trust, or the discourse could never get started. In our time, all of the sciences are much more developed, and we tend to think of "axiom" as meaning any proposition which can be "taken as read". For example, if I want to propose an extension to the theory of Special Relativity, I will probably take the Theory of Special Relativity itself as an axiom, without rehashing all of the arguments in its favour.

"An endless string of just 1s or just 0s (or the same 1 or 0 endlessly measured can contain no information)."

An endless string of 1's will convey different information to an endless string of 0's because 0 and 1 have different definitions; consequently, such strings cannot avoid conveying information, if only "the absence of 0 (or 1)".

How would your hypothesis work in a context where the presence or absence of "x" matters, for example, where "x" equals magnetic or strong or weak or gravitational force, emf, light/dark, relative proximity, etc? I think you are confusing "information" with "variability". Human sense organs require variability to function properly, but such variability is not an essential component in the definition of "information".

What you are talking about is subjectivism and objectivity. It's covered in Philosophy 101.

You might also consider the desirability of expressing your main thesis in 200 words or less, developing further details in response to subsequent comments.

A surprisingly tricky question!

One sense of A is always synonymous with B. Therefore, it will ALWAYS be the case that one sense of A is interchangeable with B. The real question is: can we imagine a context where the alternative sense of A is incompatible with B?

I'm thinking along the lines of A = the square root of 1, which as you know may be 1 or -1, but I'm not sure how to proceed from there...

I'm glad Mihai got some meaningful replies, but was this ever a question in mathematical philosophy? If the moderators had been more vigilant, they would have moved it to Ethics/morality/religion.

the ball as a ball in space is suspended. — Astrophel

Like if you're not wearing underpants?

In my view the original question is a classic example of "overthinking the problem". The proposition, "This statement is false" does not embody any significant philosophical paradox. It is merely a grammatical curiosity. The subject - "this statement" - does not contain any positive assertion; consequently, it can be neither true nor false, and the assertion of it is invalid. As a mathematician would say, the original proposition is "undefined". If "This statement" referred to some preceding statement - for example, "All chocolate is brown" - then it would have a positive content which might be true or false. Putting it another way: the proposition is self-referential and thus circular (ie invalid).

I've always found that downwards causation correlates positively with the quantity of Semillon consumed... sorry, I know I'm being facetious, but I think the philosophical implications have been fairly well explored, and it's time for a little humour.

Much of your post is not clearly intelligible without considerable further elucidation; what is intelligible, would seem to prerequire and presuppose a complete theory of knowledge. Try to refine your question to a single, focussed point of discussion.

Thank you everybody... when I posted the original question, which was really addressed to the moderators, I had no expectation of such a lengthy, interesting, and varied list of replies. I am surprised to learn that some mathematicians are inclined to be suspicious of transfinite theory, and question its value, because all of the philosophical problems of TF can be ultimately traced back, in logical sequence, to the 19th C Peano/Dedekind axioms of arithmetic - and if you are going to question those, then you can no longer assume that 1+1=2, and the entire science of mathematics falls on its butt in a crumbling heap!

As to different levels of infinity - the proofs that there must be at least two levels of infinity are so childishly simple, they can be understood even by high-school students who are not particularly bright in mathematics. The validity of the countable or aleph-null infinity is embedded in the properties of the natural number line. The validity of the aleph-1 or non-countable infinity is adequately demonstrated by Cantor's Diagonal Argument, which is very simple, and readily intellgible to non-mathematicians. Anybody who wishes to deny that higher-level infinities are valid in mathematical philosophy, must begin by pointing out where Cantor went wrong.

Are aleph-1 infinities useful in the sciences? Difficult question. We know that aleph-null infinities are not useful for investigating the origins of the universe, because every variable that we would wish to measure simply approaches infinity (or 0) as we approach closer to the singularity, and this of course is non-informative. Will the solution entail finding a way to apply the concept of the aleph-1 infinity? Some day, we'll know.

That's true. I have read that some high school teachers tell their students that (for example) infinity minus infinity, or multiplied by infinity, or added to infinity, is undefined. Serious disinformation!

By the way, I would ask, are you aware that the infinity of the real numbers between 0 and 1 is a different kind of infinity to the infinity of the natural number series? Forgive me if you already knew that; if not, I recommend you google Cantor's Diagonal Argument for the non-countability of the real numbers.

We can argue against the logical possibility of a miracle, because the definition of "miracle" given here presupposes the existence of a Godlike being; and nobody has yet succeeded in establishing the existence of any such being, beyond the level of assertion unsupported by scientific evidence.

In the history of science, there are many phenomena which could not be explained by contemporary scientific orthodoxy. In every case, the difficulty has been removed by improving the theory or by replacing it with a better one. As yet there is no reason to believe that miracles, as defined above, actually occur or are necessary to complete scientific knowledge.

The "miracle" hypothesis is, prima facie, unnecessary to science, and is chiefly of value to those who prefer the paths of darkness and superstition.

According to the Curry-Howard model - with which I admit I am completely unfamiliar - what takes the place of the axioms in mathematical science?

Well, when we explore space, we don't see any Boltzmann brains — RogueAI


Cosmology at its best. A PhD thesis could be written on this observation. :roll: — jgill

A hit! A palpable hit! (Hamlet)

How many decimal numbers are there in between any two whole numbers (such as 0 and 1)?

Infinity.



How many whole numbers are there?

Infinity.



Therefore, there are an infinite number of infinities. — an-salad

Yes. What's your question?

Bertrand Russell's own "Introduction To Mathematical Philosophy" is the text you are looking for. Written while he was serving a prison sentence as an anti-war protestor, it is essentially a summary translation of Principia Mathematica into ordinary language. It does not presuppose any knowledge of mathematics or formal logic. Just reading the first three or four chapters will transform your whole world-view.

By the way, please don't fall into the trap of supposing that Aristotelian logic is a valuable guide to truth in the modern world. By all means study it, if Classicism and Scholasticism are your special areas of interest. But Aristotelian logic in the modern scientific and philosophical world is largely irrelevant. It has long since been superseded by developments in many areas, including set theory, Peano/Dedekind arithmetic, and relational (as opposed to predicate) logic.

If anything is an appearance it is known mediately,

One assumes, through the senses?
The individual knows that he (or she) acts non-mediately

Whoa, back up the wagon, Chester! How do I know that my "action" is not just another appearance known mediately? Are we confusing "action" with "the will to action"?

Thus, action cannot be an appearance.

Non Sequitur.

An interesting question, but I am a lazy person, so without research I will just point out one common fallacy which occurs too often in Wikipedia and other forums: namely, the assertion that Euclid's Postulate 5 and the parallel postulate are logically synonymous. This is easily disproved by pointing out that a negation of the parallel postulate (within the Euclidean context) does not entail a negation of Postulate 5.

The highest concentration of fallacious arguments in one location which I have ever come across is not in a speech by a politician or reality-show imposter, but in the Wikipedia article summarising all of the supposed proofs that 0 is an even number. I have seldom encountered so dense an intellectual fog within the compass of a single essay, even from a first-year student.

Your use of the word "regard" is problematic. Do you mean to say "makes possible" or "facilitates"?

I think if you accept that A Priori knowledge is possible, then it necessarily determines experience-based knowledge, since "A Priori", by definition, is logically anterior to the data of experience, and must provide the framework within which that experience is interpreted and classified.

But you are touching on one of the fundamental problems in philosophy which is yet to be resolved. Although Kant didn't express it in quite these terms, the problem you raise is the problem of the "axiom" - the proposition which is apparently self-evident to reason, but can't be proved, and yet, which must be accepted as true if the philosophic discourse is to proceed to higher stages. The "Axiom of the parallels" is a case in point. In two and half thousand years, nobody has come up with a proof. But it underwrites almost the whole of Euclidean geometry (to the intense embarrassment of mathematicians and philosophers).

On the other hand, the monoclastic neutrinal differentiation of the autosomatically-determined spin value inherent in all such equitational bivalent transmogrifications cannot be entirely ignored, wouldn't you agree? Particularly if we keep in mind the various M&M chromatic values which may be obtained at little expense in our own time.

I'm sorry, what was the question again?

On the other hand, the common notion that God's nature is "unknowable" or in some sense "hidden" presents severe logical problems. It is obvious that if God's nature is hidden from us, then it must be impossible to prove, or even to advance any argument, that God actually exists. How can you prove that something exists, if its very existence is hidden from us? Putting it another way, if it has no properties which can be perceived by us, why would we even think it must exist?

fdrake, I think the preceding message was actually written by the author of "Christoff", Christoff being a half-baked bug-ridden AI program. Does "productionwise" tell you anything?