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 It has long since been superseded by developments in many areas, including set theory, Peano/Dedekind arithmetic, and relational (as opposed to predicate) logic.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.

"Philosophers are also humans and like to socialise and go off topic."

That is why we have Facebook and Twitter.

Unfortunately, objective analysis in philosophy does not recognise the right to go "off topic". "Off topic", by defnition, is not relevant to the question being posted. Or am I being too subtle here?

On the other hand, if you are referring to brainstorming (which can have a philosophical value), perhaps the page name should be changed to reflect this. "Shouting" is chiefly associated with Fascist and Communist philosophies.

"Any truth about our origins is relative to us"

Are you appealing to this as an axiom?

I'm afraid I find most of this thread unintelligible. My attention is fixated upon the proposition that in some way we could make 1 +1 = 2 + 1. What modification would we propose to the axioms of arithmetic to make this possible?

Or is Agent Smith subconsciously confusing sets and subsets? Because if we have a set of 2, the number of possible subsets which can be formed is equal to 2^2, or 4.

"Solidity" is directly and immediately relative to the strength and rigidity of the molecular bonds of the material in question. Molecules of iron form very strong and rigid bonds, which explains why you won't fall through a steel floor. The bonds between iron oxide molecules, on the other hand, though rigid, are quite weak, which explains why you will fall through a rusty floor. The bonds between molecules of dihydrogen oxide are incredibly flexible but very strong, which explains why you can dive into a swimming pool without hurting yourself (hopefully!).

The molecular bonds of polyethylene are extremely stong, but not very rigid, which explains why "polythene" is flexible but so difficult to tear apart.

"As a newborn, our sensations are incoherent"

This is actually quite a large assertion, requiring considerable argument. The fact that a baby is generally incapable of interacting socially in a meaningful way, whether by speech or action, does not suffice to prove that its sensations are incoherent. It proves only that its physical (and neurological) development are inadequate to express any thoughts it may be capable of articulating.

"My consciousness is all that I really know exists"

YES! If only Descartes had had the wit to understand this... but he lacked the intellectual honesty.

It is difficult to conceive Mishima's point exactly because of the difficulties of translation... I take him to mean that because German philosophy is so metaphysical, it lacks any point of contact or 'ausgleichfunktionsknopf' to connect it with the real world. I assume that the symbolism of the toilet refers to the only outlet of any dissenting world-view

But I would assume, since that time, that this has been more than countered by the influence of the "Scuola di Coca-Cola" of Western culture?

This thread can drone on as long as one likes! Until someone addresses a couple of elephants on the sofa:

(1) What is "good"? How is "goodness" to be defined?

(2) Ought we to be good? Is there such a thing as a moral imperative to be good?

Sorry to harp on the obvious like that but, as the ancients realised, you must start by defining your basic terms and, with luck, you'll find that many of the questions answer themselves.

The relevance of mathematics would be fairly obvious to anybody who had taken Philosophy 100 in Utilitiarian ethics. But please don't underestimate the philosophy of arithmetic. It is at least as problematic as the philosophy of any other branch of mathematics, precisely because it deals with the most primitive or elemental levels of the science.

Agent Smith is a philosopher; he provokes thinking, which is, of course, frowned upon in any good society.

"According to The Elijah Price/Mr. Glass Principle, at least one actual infinity exists."

I know nothing about Mr Glass or Elijah, but the thesis is provable in set theory, provided you accept that the employment of axioms is inevitable in any science. It's known as the "axiom of infinity": For any natural number n, there exists at least one set with n members. At first glance, this doesn't look as if it had much to do with infinity. But perpend.

Let's say you choose the number 9. Well, there must exist at least the set of the numbers from 1 to 9, otherwise the number 9 would have no meaning. But if you have a set of 9 members, the total number of subsets formable would be 2^9; and from those you could form a further subset, with a cardinality of 2^2^9, and they would form a further subset... and so on ad infinitum.

"God's worst ideas are our best ideas."

Before this "discussion" gets transferred to the religious forum, let me say this: there can be no continuity between our ideas and god's ideas because god possesses all of her qualities to an infinite extent; consequently, from god's perspective, human thinking and human affairs must appear to be infinitely trivial and inconsequential and, quite probably, of no logical value whatsoever.

But of course, my last does not address your main point, which I understand as follows: the existence of an event horizon implies a stronger gravitational field than any we can find in the observable universe. But we do not need to go so far as proposing a singularity of infinite mass to account for this phenomenon. The central mass would only need to be slightly larger than any we can observe. Is that your position?

Michael, your point needs a bit of unpacking. It's often said that "Black holes are... a region within which gravity is so strong that light cannot escape."

"Escape" is a tendentious word to use, because it encourages a sort of Newtonian picture of light struggling upwards for some distance and finally succumbing to the force of gravity, as a rifle bullet would on Earth.

But this is a misleading picture. Light naturally travels in a straight line but, because it has mass, it will follow the contours of a gravitational field. In the neighbourhood of a singularity, gravity curves so tightly back on itself that it creates a hard discontinuity in space/time (like the boundary of a blob of oil floating on water) which we call the event horizon. Light which is inside the event horizon continues to travel in a "straight" line, but following the contours of the gravitational field. So the space/time discontinuity prevents us from seeing it.

The natural tendency of a railway locomotive is to travel in a straight line; but, where the tracks are laid, that's where it will go!

Yes, I think there's a Windows update.