Is the statement "it is logical to conclude that this statement is not logical" logical, or isn't it? Your syllogism falls apart (and the analogy is pointless).
No, Occam's razor would tell me to use logic in science, unless logic was found wanting. Occam's razor tells us something about science; it tells us nothing about reality.

I think you mean, all mathematics that adopts logic as an axiom. People have already done mathematics that does not. (Say, the mathematics of "fuzzy logic", for example.)
And who did the expelling, logic or Russell? You are admitting that logic is a servant not a master, yet before you claimed the universe was logical. What this is is an example of the role of consensus in mathematics, and it has its role in science too-- but it does not define objectivity.
I am well aware that many aspects of mathematics were built to conform to the real world. I consider those aspects to be science, for just that reason. It all hinges on the authority-- when the axioms are their own authority, one is doing pure mathematics, and when familiarities with reality are sovereign over the axioms, it is science-- expressly because only the latter requires the separation of subject (the mathematician) and object (their experiences with reality). Where is Hardy when you need him?

But the use of all these concepts in mathematics is quite different than in science. They all refer to how uncertainties in the domain space map, precisely, into uncertainties in the image space. For example, when mathematics uses probability, it is over a known distribution. Science never knows that, yet must function anyway by modeling it. Again, science borrows from the tool of mathematics, but the tool is precise-- it's usage (science) is imprecise. You've said the same yourself, about the absence of inconsistencies in mathematics. Science is full of them, we dance among them so automatically we almost forget we are doing it.
We can agree that logic works extremely well in the context of objectivity, and not terribly well in things that are more clearly subjective. So logic dovetails closely with objectivity. Indeed, one can say that it is logical to use the subject/object separation expressly because of its scientific successes, which is a way of saying that mathematical thinking supports scientific thinking. We can also say that scientific thinking motivates mathematical thinking. It is probably a chicken-and-egg situation. I'm just saying that the logic and the objectivity are defined separately, like the different fingers that hold the pen-- that they work so well together is not a reason to equate them, and indeed I feel a lot of the misconceptions about what science is (by scientists) stem from confusing it with what mathematics is.
Yes, I'm a taker on (a), but I would not say "logic is objective", because it is not defined in the same way, but I would agree that "logic works well on objective projections of reality", which is pretty much what I hear you as saying here. Why it works, I have no idea, and it's tempting to say it is "because it is itself objective", but we don't really know that. We might be pretending away a very significant mystery.

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Physics doesn't predict the future, it predicts the past that hasn't happened yet.

There are two kinds of delusions, the obvious kind that clearly don't work, and the insidious kind that clearly do.

Alleged inconsistency #1:
Solution: You've committed the fallacy of excluded middle. There is a third possibility: that the statement makes sense, but is false. This possibility leads to no paradoxes.

Alleged inconsistency #2:
Solution: That's a straw man. Quantum mechanics would never be necessary, because the body of the turtle and the feet of Achilles are macroscopic objects, not subatomic ones.

Alleged inconsistency #3:
Solution: Fallacy of equivocation. I have never claimed that "life and death", and pain and happiness, etc., were governed by logic. When I used the word logic, I was obviously referring to physics, chemistry, biology, and so on. Not everything we deal with in life is governed by logic, but everything that science deals with is. This conversation is about science.

Alleged inconsistency #4:
Solution: there is no middle. The empirical evidence weights heavily against the notion that logic leads to no results.

Alleged inconsistency #5:
Solution: Another fallacy of equivocation. The real numbers are so named because they represent all the sizes that a physical object can have. The complex numbers, while useful in some contexts, do not represent, by themselves, measurements we can make in the real world. While their real and imaginary parts may have physical interpretations in some contexts, the real part and the imaginary part are both real numbers.

Alleged inconsistency #6:
Solution: You've committed the fallacy of begging the question. You're treating the entity you've defined as though it were a set, but you never showed that it was one. It isn't. As Gödel notes, one should not define sets by subdividing the abstract collection of all sets, but rather construct them from well known objects (such as the natural numbers).

This was fun. Let me know when you've come up with that example of an illogical universe I asked you to produce.
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"A paradox may be paradoctored."

Thank you for that very concise summary - it seems I missed the emphasis that Disinfo Agent was placing on the value of consensus as a defining characteristic of objectivity. I was getting confused over the agreement in this discussion with regards to the use of objectivity in science and how we attempt to use objectivity despite the underlying impossibility of separating the subject and object. But the crucial disagreement is, as you rightly say, related to the meaning of that word which very importantly defines scientific knowledge. It is an important question.

Wrong. It makes no difference at all what other possibilities exist, so there is no "excluded middle" fallacy. The statement is all about whether or not it can be logically reasoned to be true, period. You are essentially admitting that logically reasoning the statement is true leads to a contradiction, so since you think reality is logical, this implies logically that the statement cannot be true. This further means that we have logically argued the statement cannot be true, so again if reality is logical, it therefore cannot be logically argued to be true-- and that's all the statement asserts. Ergo, we logically reason the statement is true, and you have not escaped the paradox by introducing a middle possibility that I neither excluded nor even needed to consider.

Again you simply do not escape the problem. My whole point is that quantum mechanics would be necessary to answer the question "please test the moment when Achilles catches the turtle to 20 decimal places of accuracy", because to test small enough time intervals to achieve that, you will come smack into quantum mechanical limitations (the uncertainty principle). The truth is, there is simply no way, even in principle, to satisfy that requirement-- the universe itself does not seem to let us ask that question. Nevertheless, Achilles clearly goes from being behind the turtle, to being ahead of it, even though the universe doesn't let us ask when he pulled even to a precision greater than some limit. That's the kind of "fuzziness" that we have to address in physics all the time, or more correctly, we simply choose not to address because we don't need to-- unless we arbitrarily require the universe to be logical.

That is also false. You used the word "reality". So your position is only consistent if you do not think pain and happiness, etc., have a reality that is not covered by chemistry, biology, and so forth. You are welcome to believe what you like, but don't expect nonscientists to have any need to agree-- or scientists either.
Actually, the distinction you are making now is one that I have labored quite long to establish-- and now you pretend it was your position all along? Need I remind you of this criticism you lodged of my quote (my quote in italics):
Are you backtracking on that now? I just want to understand the current state of your argument.

But you just cited some cases above when it led to no results. Are you claiming that it always leads to a result in science, but magically not in other pursuits? There is a middle in other things, but there cannot be in science? If so, what is the basis of that assumption-- I've given examples where logic leads to no results in perfectly scientific situations, like asking for a logical analysis of the moment when Achilles catches the tortoise.

What makes you think this? It's completely false. On what basis do you claim a physical object can have a circumference that is exactly pi? It can't even meaningfully be said to have a circumference of 3.14159265358979323846... because that much precision is meaningless for physical objects. Indeed that is the point I'm making, which you think is equivocation but in fact is the simple truth established by science.
No, what I've done is examined a ramification of your claim that scientific reality is logical, never fuzzy. Scientific reality involves analyzing sets, often infinite ones, like the integers-- and note that the set I created was constructed, it was not generated via any subdivisions. So if mathematics/science/reality are all one nice neat logical bundle, as you have often claimed, then what I am doing is completely valid and begging of no questions whatsoever. Another miss.


All right, once again I will rise to the challenge. Construct, mentally if you will (as logic dictates we be allowed to do), the set of all natural numbers. That's a construction. Now run through them all, pairwise, creating new sets of two integers, a la Cantor. Now run through all triples, etc., now continue over all integers. Each time, we include the set we have constructed only if it does not contain itself. Group together all the results-- we have a construction of the set of all sets of integers, that further do not contain themself, yes? Now let us ask a simple question-- does that set contain itself? Do you know?

I'm sorry, but I have you batting 0/6 in these "solutions". I conclude this "completely logical reality" you imagine is nothing but a nice warm cocoon of self-assurance, but the evidence is to the contrary.

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Physics doesn't predict the future, it predicts the past that hasn't happened yet.

There are two kinds of delusions, the obvious kind that clearly don't work, and the insidious kind that clearly do.

Yes, contrary to what some might think, I don't actually like to spend gobs of time on pointless flights of fancy-- I do view these as crucial issues, and very important to have straight before looking at deeper concerns like interpretations of quantum mechanics or when the authority of science is mandated versus optional.

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Physics doesn't predict the future, it predicts the past that hasn't happened yet.

There are two kinds of delusions, the obvious kind that clearly don't work, and the insidious kind that clearly do.

I will address only Ken's most interesting objection, for now.

When you say sets of "pairs", "triples", etc., I assume you actually mean the subsets of N with cardinality 2, the subsets of N with cardinality 3, and so on, and not ordered pairs, triples, etc., which is what these words normally mean in mathematics. And by "set S contains x" I assume you mean that x is an element of S.

In the formalization of the integers that I'm familiar with, which as far as I know is the standard one, {{1, 2}} is different from {1, 2}. Therefore, the entity you have defined is indeed the power set of N, and like any set in the modern standard formalization of set theory, it is not an element of itself.

You're very fond of claming victory before the battle is over. It ain't over till the fat lady sings.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Since my use of the word "consensus" seems to have worried some people (even though I actually used the terms "agreement" and "collaborative" first), no doubt because it conjures up images of like-minded people getting together in clubs to confirm each other's prejudices -- which was not at all what I had in mind --, I am also replying to the following:

Many fields aspire to consensus, but wishing for something does not make it so. Science is the only human endeavour that has repeatedly shown that it converges towards a consensus as it develops. No other human activity can claim the same.

I would add that science is the only human activity which is fully objective (objective contributions can also be useful to other human activities, but do not define them). Since I feel, as I've been arguing, that disciplines such as quantum mechanics, psychology, sociology, or mathematics lead to objective results, I see no reason to exclude any of them from the list of the sciences.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Yes, none of the individual sets I am creating will contain themself, I am well aware of that. Now take those sets and make all the sets that can be made with them, and then lump all those together into one set that contains all the sets you can make with integers and sets of integers, with the further property that no set in that uber-set will contain itself. My question remains unanswered: does that set contain itself?

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Physics doesn't predict the future, it predicts the past that hasn't happened yet.

There are two kinds of delusions, the obvious kind that clearly don't work, and the insidious kind that clearly do.

I think it is fair to say there is a widespread "consensus" that murder in cold blood is wrong, yes? Certainly there are more people who accept that than accept the postulates of quantum mechanics! Shall we say that "murder is wrong" is an objective statement based on your definition? Shall we devise a scientific experiment to establish that murder is wrong, as science is the "only objective human endeavor"? No, because "wrongness" is not an "object", no matter how much consensus it enjoys.
There is no human activity that is "fully objective", by either definition-- it is merely an idealization of what humans are capable of.

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Physics doesn't predict the future, it predicts the past that hasn't happened yet.

There are two kinds of delusions, the obvious kind that clearly don't work, and the insidious kind that clearly do.

"Made with them" in what way?

You're being too vague. That's the basic problem with all the examples you've come up with. They're based on manipulating very vague notions as though that were unproblematic. But it's not unproblematic, it leads to inconsistencies like Russell's paradoxes. In mathematics, we have learned that we must be very careful and precise in how we define sets. Naive, amateurish set manipulations are just like naive, amateurish manipulations of infinite series: until you realise that you need to check for convergence before you can treat a series like an ordinary sum, you're walking on thin ice.

You can't use naive, outdated mathematical paradoxes, and ignore all the developments that have happened since those paradoxes were discovered, to claim that mathematics is irreparably inconsistent. That would be like me confronting 19th century classical mechanics with electromagnetism, to claim that theoretical physics is gibberish. It's not, the inconsistencies between Newtonian mechanics and Maxwell's electromagnetism were overcome by Einstein.

As further study improves our understanding, the apparent contradictions unravel.

That statement is tautological, because it's of a legal nature. "Murder" is the name of a crime, and crimes are by definition acts which a society regards as wrong. When you instead ask whether killing is always wrong, or how a murder in cold blood should be punished, the alleged consensus disappears.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Watching this thread is better then a movie. You guys are fantastic. [/offtopic]

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I haven't really been keeping up with the conversation, but the way I see it, mathematics is a tool for demonstrating truth, and when used correctly, it is very objective in what it does. "Correctly" in this case means, applying a formula that actually works and, say, not messing up and forgetting to carry the 2.

If I have one apple, and get another apple, therefore I have two apples. This equation will always hold true. Nothing can change it; there is no way I can have 1 apple + 1 apple and end up with 3. Even with super science, I need to add another apple into the equation to get it.

I suggest that if something seems to disobey mathematics, that there's something there that we don't quite understand yet and can't quite plug into the formula.

Then again, mathematics really doesn't quite seem applicable to every single little thing. But then, I was going to suggest an example using color, but really there is mathematics to colors, in the form of wavelength...

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"A long habit of not thinking a thing wrong gives it a superficial appearance of being right." -- Thomas Paine

Being intelligent is not a felony. But most societies evaluate it as at least a misdemeanor. -- Heinlein

Creationists make it sound as though a "theory" is something you dreamt up after being drunk all night. -- Isaac Asimov

Yes, but math and its application to nature are not the same thing. You can have powerful mathematical constructs that have no application in nature.

Math has no idea on its own whether or not these apples are reproducive, for instance. For all it knows, your first apple could be pregnant and produce a 2nd apple, though of a small size. Now 1 apple plus nothing yields 2 apples. Our new math arguement will be that we should see 4 apples by simply waiting. Science steps in and looks for the math connections to nature, as Ken has stated, deciding whether the path illuminated by math is a realistic path or not, using my analogy.

... and the number of photons per second. Have we talked about the Sun's color lately?

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Lighten up! This is a stellar board! Author: duh.

"The Sun, with all the planets revolving around it, and depending on it, can still ripen a bunch of grapes as though it had nothing else in the universe to do..." Author: Galileo supposedly.