More/Less Mathematics

[Martin_]

I have never had a problem multiplying 2 times pi.

Perhaps you can then share the correct answer with us mortals?

2pi?

What about some randomly picked binary ZFC real numbers, can your divine omniscience also tell does the sum of 0,000... + 0,000... start with 1 or 0?

Now it's getting interesting. I've seen these things before, 1=0.9999999... etc.

Please show me why the obvious answer "0" is not as obvious as it seems.

[SanteriSatama]

What about some randomly picked binary ZFC real numbers, can your divine omniscience also tell does the sum of 0,000... + 0,000... start with 1 or 0?

Now it's getting interesting. I've seen these things before, 1=0.9999999... etc.

Please show me why the obvious answer "0" is not as obvious as it seems.

The problem is, nobody can show,. That's what "Axiom of Choice" means in practice, it postulates non-demonstrable "real numbers" which don't even have any generative algorithm. Only Omniscient God could tell, if either number contains a "1" at some point, perhaps few galaxies away, or in the distance of infinity of Aions. And I also reject theology of omniscience as mathematically inconsistent.

So, carry rules of addition become here pure guesswork of ???

[ScottRoberts]

I have never had a problem multiplying 2 times pi.

Perhaps you can then share the correct answer with us mortals?

2pi.

As for the approximations mortals get by punching some buttons of their calculators, they have nothing to do with set theory and real numbers. Floating point arithmetic is rational arithmetic.

Yes. What does this have to do with what we are discussing? One doesn't do set-theory-based group theory on calculators.

What about some randomly picked binary ZFC real numbers, can your divine omniscience also tell does the sum of 0,000... + 0,000... start with 1 or 0?

Not sure what you are getting at here. Do you mean that since I don't know what is at the fourth decimal place I can't be sure what the third decimal place of the sum will be? If so, yes, I can't tell. Why is that a problem? In any case, one doesn't deal with "random real numbers". In practice one just deals with those that one can work out their decimal expansion to whatever precision is needed (for the application). If there is no physical world application involved one just continues to use the notation (pi or sqrt(2) or whatever). Since this property of being able to work out the desired decimal expansion is preserved by arithmetic operations, one can do arithmetic with such irrationals. Indeed one should do so (some may cancel).

As to the Axiom of Choice, you will note that abstract algebra textbooks are careful to say "if the AoC then Zorn's Lemma", not "since the AoC then Zorn's Lemma".

[SanteriSatama]

I have never had a problem multiplying 2 times pi.

Perhaps you can then share the correct answer with us mortals?

2pi.

That's not multiplication. No arithmetic operation has been done, despite the claim.

The "axiom" that real numbers form a field is not an axiom. It's logically dishonest claim. It violates the basic syllogism requiring that members of a set really can do what the set is claimed to be able to do. By violating that syllogism, there's no logical continuity.

Without property of a field, and hence ring etc., there's no logical continuum to group theory. In it's current form of real number sophistry, group theory can't prove anything, it has no theorems, only conjectures. How ever deep and meaningful those conjectures appear, they are mathematical speculation derived from an 'if-then' form, where the "if" part is not a logically acceptable proposition, but fraudulent sophistry.

I'm sorry, maybe I'm just being emotional, but I do feel that mathematics should not be deceptive sophistry. Proofs, demonstrable proofs instead of declarations of existence, should still matter, despite their incompleteness in the formalist if-then speculations. No blatantly inconsistent proposition should be accepted as the "if" part of a formalist speculation, while it is also claimed that consistency still matters.

My own background is translation. In our field, loyalty to the text is everything, even though no translation can ever be exhaustive and complete final translation. We have our professional ethics, as people need to trust that despite our limitations, we do our best to convey meaning and serve communication.

From translator point of view, ethics should still matter when speaking and writing etc. communicating mathematics. Ethics of honest language should matter also and especially when communicating with the physical spirit/potential of quantum superposition, as language of mathematics is how physics measures/decoheres/translates actual from potential.

PS: on the p-adic side, things are not as bad as on the real side. Carry rules work much better:




P-adics are in a sense 'top-down'.

[ScottRoberts]

2pi.

That's not multiplication. No arithmetic operation has been done, despite the claim.

According to your definition of arithmetic. It is multiplication according to mine.

The "axiom" that real numbers form a field is not an axiom. It's logically dishonest claim. It violates the basic syllogism requiring that members of a set really can do what the set is claimed to be able to do. By violating that syllogism, there's no logical continuity.

If + and x are defined for real numbers, then the real numbers are a field. If not, not. So it is again just a choice of which mathematics one wants to work with.

Without property of a field, and hence ring etc., there's no logical continuum to group theory. In it's current form of real number sophistry, group theory can't prove anything, it has no theorems, only conjectures.

The whole point of group theory is to provide theorems that are true of all groups. If one doesn't want to include the real numbers as a group, then, again, that's your choice. My only point of bringing it up in re calculators was to say that what is valid mathematics does not depend on what one can calculate on a calculator. But of course a much more direct way of saying that is to note that only a finite number of rationals can be displayed on a calculator. So the fact that π can't be does not mean anything.

How ever deep and meaningful those conjectures appear, they are mathematical speculation derived from an 'if-then' form, where the "if" part is not a logically acceptable proposition, but fraudulent sophistry.

All mathematical proofs are "if-then" constructions. So one's mathematics will depend on what one finds logically acceptable on the "if" side. Which, again, boils down to a choice. In the case of the AoC there are those who consider it unacceptable, and those who say "maybe, maybe not, but if ok, then we get these results". In either case, one is doing legitimate mathematics.

I'm sorry, maybe I'm just being emotional, but I do feel that mathematics should not be deceptive sophistry. Proofs, demonstrable proofs instead of declarations of existence, should still matter, despite their incompleteness in the formalist if-then speculations. No blatantly inconsistent proposition should be accepted as the "if" part of a formalist speculation, while it is also claimed that consistency still matters.

A system is inconsistent if both A and ~A can be proved. I am unaware of this happening in the 150 years or so that mathematicians have been working with Dedekind-cut-defined real numbers.

PS: on the p-adic side, things are not as bad as on the real side. Carry rules work much better:

Go for it. When you can produce the equivalent of the Euler equation, get back to me.

[SanteriSatama]

If + and x are defined for real numbers, then the real numbers are a field. If not, not. So it is again just a choice of which mathematics one wants to work with.

Well, are + and x defined for all members of the set of real numbers? If they are not, can you honestly claim that they are consistent and don't violate the most basic syllogism?

All mathematical proofs are "if-then" constructions.

All? Are Zeno's paradoxes "if-then" constructions? The proof by absurdity (that continua are nor reducible and hence more fundamental than discreta, main argument of Eleans) is derived from intuitive and empirical evidence, not from any if-then construction as such.

So one's mathematics will depend on what one finds logically acceptable on the "if" side. Which, again, boils down to a choice. In the case of the AoC there are those who consider it unacceptable, and those who say "maybe, maybe not, but if ok, then we get these results". In either case, one is doing legitimate mathematics.

Do you really mean that logical acceptability is purely a matter of choice, and anything goes? Gödel doesn't matter? Constructions can deviate from their axioms as they like, when axioms are only implied and not explicated? It's OK to apply LNC only if and when and how one likes, and conveniently forget it when it feels like to choose so?

A system is inconsistent if both A and ~A can be proved. I am unaware of this happening in the 150 years or so that mathematicians have been working with Dedekind-cut-defined real numbers.

AFAIK nobody really works with Dedekind-cut-definitions, after the cherry picked text book example of square root of 2. So is anyone really looking into that specific definition?

Maybe that's what Conway did with Surreal numbers? The most (and only?) consistent construction that can be done with a Dedekind cut? Well, it has game theoretical beauty, but you can generate and find 1/3 only after infinite round of games. Which is not exactly an improvement to Stern-Brocot tree(s), but still, an interesting and different point of view from the speculative and heuristic formalist side of games.

Go for it. When you can produce the equivalent of the Euler equation, get back to me.

Which identity do you mean? The one with the transcendentals, or the doubly infinite series?

On a general note, I do confess to the view that mathematics should be... not "unity", but still a coherent whole/process. Even though I sometimes joke in the Derridan post-structuralist lingo for my self-amusement, I do not support in long term the post-modern condition of fragmenting mathematics etc. into arbitrary language games, which stop talking to each other in constructive manner.

If set theory is/was a chrysalis, maybe some day...

[ScottRoberts]

Well, are + and x defined for all members of the set of real numbers?

Yes. See any textbook on Real Analysis for the proofs.

All? Are Zeno's paradoxes "if-then" constructions? The proof by absurdity (that continua are nor reducible and hence more fundamental than discreta, main argument of Eleans) is derived from intuitive and empirical evidence, not from any if-then construction as such.

The Eleatics made their arguments before Euclid established what a mathematical proof is.

Do you really mean that logical acceptability is purely a matter of choice, and anything goes? Gödel doesn't matter? Constructions can deviate from their axioms as they like, when axioms are only implied and not explicated? It's OK to apply LNC only if and when and how one likes, and conveniently forget it when it feels like to choose so?

Anything goes as long as what one gets doesn't degenerate. A system that produces a violation of the LNC immediately degenerates.

AFAIK nobody really works with Dedekind-cut-definitions,

Of course they don't, no more than one thinks through a proof of the Pythagorean Theorem when calculating the length of a hypotenuse. One learns all about Dedekind cuts in a course on Real Analysis, and so gains confidence that arithmetic on reals works, and then ignores it as one merrily goes on to multiply 2 times π.

Go for it. When you can produce the equivalent of the Euler equation, get back to me.

Which identity do you mean? The one with the transcendentals, or the doubly infinite series?

e^(iπ) + 1 = 0

[SanteriSatama]

Anything goes as long as what one gets doesn't degenerate. A system that produces a violation of the LNC immediately degenerates.

What do you think of this, in terms of degeneration? It follows from the axioms of ZFC that empty set is pairwise disjoint. On the other hand, the axioms forbid writing a set of disjoint empty set's, ie. {{}{}} is not allowed.

A formal consequence of the set of axioms is forbidden by the same set of axioms. How does this relate to LNC?

One learns all about Dedekind cuts in a course on Real Analysis, and so gains confidence that arithmetic on reals works, and then ignores it as one merrily goes on to multiply 2 times π.

In his video lectures, prof. Wildberger goes into great depth and length to show that arithmetic on reals does not work under any definition, and that in fact, there is no satisfying definition of reals in any standard text book. The whole discussion is too wide and detailed to repeat here, but if you are interest, and want to hear and see the criticism, maybe you are able to see some errors in his line of argumentation, or not.

A related comment on the group theory. It's my impression that Wildberger, who used to work on Lie groups, ran into deep troubles in that area, because of weakness and restrictions of the standard frame of set theory reals, and that motivated him to start from scratch with his construction from computable foundation. Coming from background of Lie groups, and losing faith in that area, it does not sound surprising that he considers continuum the greatest unsolved problem in mathematics.

Which identity do you mean? The one with the transcendentals, or the doubly infinite series?

e^(iπ) + 1 = 0

I like the shorter form better:

e^(iπ) = -1

Finnish word pieni, containing pi, e, n and i, means 'small'. When I told about this little finding of mine to my mathematician father during his last days, he was very pleased.

[ScottRoberts]

What do you think of this, in terms of degeneration? It follows from the axioms of ZFC that empty set is pairwise disjoint. On the other hand, the axioms forbid writing a set of disjoint empty set's, ie. {{}{}} is not allowed.

A formal consequence of the set of axioms is forbidden by the same set of axioms. How does this relate to LNC?

I can't comment without seeing how it follows (and then -- given my being out of the game for decades -- there is the question of whether I will be able to follow...) But suppose it is a valid deduction, it could be that some tweak to the axioms could remove the problem. Or use some other way to continue to use infinite sets without it generating Russell's Paradox.

By no means am I saying that ZFC, or even just ZF, is forever going to be considered a valid foundation. After all, it is known that no axiom system of sufficient complexity to be interesting can be proven to be consistent. What I do know is that most college-level math textbooks (at least they did in my time) start out with an overview of naive set theory (nevermind ZFC) and build on that to do what they do. Maybe 100 years from now, all these textbooks will be considered to be faulty and mathematicians will be trained to work without actually infinite sets, and set theory will be relegated to the dustbin of history. I'm fine with that. But meanwhile, I decline to call all those textbook writers and those who use the results of those textbooks dishonest or to be propping up colonialism or even bad metaphysics.

In his video lectures, prof. Wildberger goes into great depth and length to show that arithmetic on reals does not work under any definition, and that in fact, there is no satisfying definition of reals in any standard text book. The whole discussion is too wide and detailed to repeat here, but if you are interest, and want to hear and see the criticism, maybe you are able to see some errors in his line of argumentation, or not.

As I said way back when, I am too lazy to do so. It's a problem, if it is one, for the professional mathematical community to deal with.

A related comment on the group theory. It's my impression that Wildberger, who used to work on Lie groups, ran into deep troubles in that area, because of weakness and restrictions of the standard frame of set theory reals, and that motivated him to start from scratch with his construction from computable foundation. Coming from background of Lie groups, and losing faith in that area, it does not sound surprising that he considers continuum the greatest unsolved problem in mathematics.

I would think the same thing could be said of Lie groups than was said of the real numbers being a field. By one reckoning, they aren't actually groups, by another they are.

I like the shorter form better:

e^(iπ) = -1

Finnish word pieni, containing pi, e, n and i, means 'small'. When I told about this little finding of mine to my mathematician father during his last days, he was very pleased.

Cute. The English word 'pie' comes close, with the 'i' doing double duty, or maybe "pie-i'd" as a homonym substitution for "pie-eyed".

[SanteriSatama]

What do you think of this, in terms of degeneration? It follows from the axioms of ZFC that empty set is pairwise disjoint. On the other hand, the axioms forbid writing a set of disjoint empty set's, ie. {{}{}} is not allowed.

A formal consequence of the set of axioms is forbidden by the same set of axioms. How does this relate to LNC?

I can't comment without seeing how it follows (and then -- given my being out of the game for decades -- there is the question of whether I will be able to follow...) But suppose it is a valid deduction, it could be that some tweak to the axioms could remove the problem. Or use some other way to continue to use infinite sets without it generating Russell's Paradox.

https://math.stackexchange.com/question ... =1&lq=1%29

The answer presupposes LEM, as does Hilbert and his followers.