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?

Introbert, you want but something to be a philosopher!

Oh for heaven's sake, people, where did you all leave your common sense? There are at least THREE logical possiblities: true, false, or "unknown/unproven/unprovable". Is 2 an even number? True. Is 3 an even number? False. Is the Parallel Postulate true? Unknown, unproven. Forgive me if I have overlooked the Law of the Exploded Sausage.

The assertion "We know we act directly and unconditionally" is, of course, fallacious; there is no reason to suppose that our action is not just another appearance, and thus known only conditionally.

But there is a deeper problem. "Conditionally" entails that there exist some criterion which is "unconditional", otherwise "conditionally" would have no meaning. Comments?

Jack, your enquiry covers an enormous amount of philosophical ground. As a starting point, pls google "natural number" and "axioms of arithmetic"!

Actually, moderators, you ought to move this question to metaphysics or ontology.

It occurs to me, someone might be tempted to object to discussion point 1 because, although we posit the space to be infinitely large, we also posit the number of B Brains to be infinitely large. But, since a B brain would not be subject to the conditions which normally restrict the development of a mammal brain, it is conceivable that it would occupy an infinitely small space.

bongo fury, please don't fall into the habit of relying on Wikipedia to support your arguments! Its contributors are, by and large, utterly unfamiliar with the philosophy of mathematics. Its article listing the arguments to prove that 0 is an even number is an intellectual fog from beginning to end.

How would you calculate density for a infinite number of things (e.g., Boltzmann brains) in an infinitely large space?

Discussion points:

1: it would be impossible. If the space is infinite then, no matter how much space you surveyed, there would always be more space beyond it, and you would have no reason to suppose its density must be consistent with the space already surveyed.

2: How are you defining "density"? Do you mean the average number of Boltzmann brains within a given cubic area? But how could anyone possibly know the answer to that question?

3: What are the physical characteristics of a Boltzmann brain? Obviously, it won't resemble anything that we routinely understand by the term "brain". Would it resemble something like a Hoyle Black Cloud?

4: Suppose there were only one Boltzmann brain, and supposing further that its physical dimensions could be quantified, then the average density in an infinite space would be 1/∞, or pretty much 0.

"P1: If circumstances change a belief, then one lacks a sufficient reason for that belief"

Obviously if circumstances change, then a belief may have to change to keep up with circumstances. That's a no-brainer; it's called "learning".

"Religious beliefs are changeable based on circumstances."

Ok, now you're joking, right?

Thund3r makes a valiant attempt to rationalise an initial post which is essentially unintelligible; what, for example, is the logical connection between natural selection and "evil"?

"For example, there are many cases of people having personal struggles which turn them to God, which is significant evidence for theism. — Ishika" - This is a joke, right?

Have you ever considered the advantages of the objective over the subjective in the advancement of knowledge?

You are at the mercy of the subjective judgements of the mod team, both individually and collectively, re low quality. We don't require your individual approval or any set of external objective criteria to establish our right to pass judgement, — Baden

You mean like in Russia and China?

OK, I think I'm getting a handle on this thread now. I thought we were discussing a problem in arithmetic where, as everybody knows, it is impossible to devise an argument that 1+1=2+1 without altering the axioms or redefining the inductive number line. But it appears that we are really discussing a problem in metaphysics, where of course anything is possible.

Hint: drop the sunflower seeds and Fibonacci numbers. Not helping.

I've noticed a sudden influx of posts in more than one forum which are peculiarly Existentialist in flavour. I have three hypotheses to explain this:

1. It's just a part of the natural flow of any philosophy forum.

2. There is a sudden flood of posts by Sartrean Existentialists who are experiencing philosophical menopause.

3. The posts are generated by AI.

How AI can satirise Existentialist philosophy to perfection!

"If there were not suitable places for such activity, it would happen more in the philosophical threads where it really does not belong."

This is essentially an admission of incompetence on the part of the moderators.