Is "1+1" provable?

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#63
vdi --

You can define the value 1 to be the cardinality of the set containing the empty set. The axiomatic things here are "set" and "empty set". This is basically all you need to construct all the natural numbers, then the integers, then the rationals, then the reals, then complex/imaginary numbers etc. etc.

to the philosphers --

Try this fact on for size: something can be both deterministic and unpredictable. What are the implications of this statement?
 
#65
MRT, etc.

This seems like a very interesting paper. I have a friend who is getting her PhD in philosphy...this seems like a good excuse to strike up a conversation with her.

They use the phrase "functional isomorphism" that is interesting because it seems similar to algebraic group isomorphisms, but appears to be weaker. I wonder how "multiple realizability" is related to multi-sorted algebra which is used in computer science to describe types, which may correspond to the notion of "kinds." I'd have to dig deeper in both of these fields, and I'm unfortunately busy right now. Mathematicians also have access to other interesting tools, such as homomorphisms, which might address the mapping from mainframe to calculator.

On a side note, aside from this just posted talk on *morphisms, we haven't actually been talking about algebra (despite some people's abuse of the term). We've brought up set theory, logic, philosophy and arithmetic, but we haven't constructed any algebras yet.

Christer --

I don't get you. First you start this thread. Next, you claim my construction of the natural numbers is bad (it isn't) because it is possible to construct infinity + 1. Then you point out the fallacy (or rather, the obvious existence of one) in the 1 = 0 argument by proclaiming: "one must be careful with infinities." And you get some facts mixed up about countable infinities. I'm confused about your intents and qualifications.

BTW, the study of calculus has been made theoretical and abstract in the branch of mathematics known as analysis.

-Won
 
#68
Re: MRT, etc.

Won said:
Christer --

I don't get you. First you start this thread. Next, you claim my construction of the natural numbers is bad (it isn't) because it is possible to construct infinity + 1. Then you point out the fallacy (or rather, the obvious existence of one) in the 1 = 0 argument by proclaiming: "one must be careful with infinities." And you get some facts mixed up about countable infinities. I'm confused about your intents and qualifications.
-Won
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First, I stated clearly why I started the thread. I cannot be held
responsible for other people starting to discuss other issues here.

I have not claimed your construction of the natural numbers to
be wrong or bad. Perhaps I made some sloppy formulation so
it could be interpreted that way. If so, I am sorry, since that
was not my intention. I rather meant the opposite, that by
going from just the Peano axioms to the set-theoretic model
you gain things like being able to construct "infinity+1" (that
is a sloppy simplification) which is used in semantics for
logic programming for instance.

Millwoods wrong proof was a joke, I presume, but it is a common
fallacy for people to do such errors if they don't understand
what they are doing when they use infinite series. The wrong
proof is a good example of how it can go wrong if you are not
careful. In fact, I think I have seen that particular example used
several times in the literarture as a pedagogical example to
illustrate this point. I don't know what you found problematic
or misunderstood, or what I misunderstood, perhaps?

Could you please point out what I got wrong about countably
infinite sets? I may very well have made some error, but since
I obviously didn't spot it myself, please tell me.

As for my intent, I started the thread for the reasons stated in
the first post. Then I have stayed in some of the discussion of
the other issues people brought up since I have at least
some knowledge of it. I am not a mathematician, though, just
a computer scientist who use some parts of logic and discrete
mathematics for my professional toolbox, so I don't want to
pretend being an expert on these issues.
 
#69
Won said:
vdi --

You can define the value 1 to be the cardinality of the set containing the empty set. The axiomatic things here are "set" and "empty set". This is basically all you need to construct all the natural numbers, then the integers, then the rationals, then the reals, then complex/imaginary numbers etc. etc.
Click to expand...

Since you complained about me being unclear or lacking
knowledge, you get me puzzled here so I wonder if you
know clearly what you are talking about. Of course we can
define the number 1 in that way, if we wish, and we can
define any integer in a similar way. We can even define,
any finite number of integers in a similar way, but could we
define them all? I may be wrong, but I think not, since we
cannot make an infinte number of definitions. On the other
hand we can construct any integer in a systematic way based
on just the definition of the number zero, as the empty set or
just as itself, and the Peano axioms. The interesting question
is perhaps are you discussing semantics or syntax, ie. do you
mean the number 1 or the language symbol "1" commonly used
to denote this number (or any other symbol we choose to
use to define the number 1, for that matter). One may also
ask if vid's original question was about syntax or semantics, but
since he admitted to not knowing much math I don't expect
him to know that himself.
 
#70
eStatic said:
And Hi Christer, please be aware that I am shooting from an aging hip here.
Click to expand...

I haven't touched most of the stuff I am talking about for quite
a while either, so I am rusty and may remember things wrong.
I also hope you didn't get the impression I pretended to be
an expert on these things, I did not mean that. I have a feeling
you know much more than me about the philosophy of science
and, at least some branches of, mathematics.



I confess that it has been a long time since I examined the details of the scientific method. But it seems to me that the assertion "...mathematics is useful for cacluating things about the real world..." is a tested and proven theory within the framework of science regardless of the fact that mathematics can and often does exist for its own sweet self. Please correct me if I'm wrong, or have missed the point here.
Click to expand...

Tested, yes quite a lot. Proven, I suppose different people will
have different opinion on whether we have proof or not or
whether we even can prove it.

I was perhaps typing faster than I was thinking, but I guess I
was also considering whether standard calculus works
for the real world models when we stretch it to the extreme.
I mean things like going arbitrarily close to zero or to infinity,
for instance. It works in the physical models we have, but does
it say anything about the real world in the extreme cases?
However, I suppose that is rather the problem that we are
stretching the physical models too far, rathern a problem
with the mathematics itself.


As a counter example consider Hilbert Space. It was, if I recall correctly, developed as a purely abstract/aesthetic endeavor and thought to have no correlation to reality. Since the rise of quantum mechanics it has been shown to be formally equivalent to the Heisenberg uncertainty matrices and the Schrodinger wave equation. I also have the impression that it has become the formalism of choice for practitioners of QM. And, if true, all this is surely passing strange. :xeye:
Click to expand...

I, wrongly it seems, assumed you were referring to "standard
calculus", ie. what developed as a mathematical tool for the
evolving physical models of the world.

There are plenty of examples of mathematical concepts which
were purely theoretical and later found useful. Much of discrete
mathematics was considered very theoretical and abstract
until it was found very useful for certain things like computer
science, and it seems a lot various branches of sciences are
starting to get interested in it. An even more striking example
is the Fourier transform, which Fourier himself called "a beutiful
theory with no practical use whatsoever". How wrong he was!



Questioning my thoughts regarding physical models: Is it not true that once one moves away from the realm where Newtonian mechanics applies, that is, into the realm of un-intuitive physics, models of existence are primarily or entirely mathematical? I.E. many of the crucial concepts require, in order to be persuasive and complete, mathematical expressions consistent with their defining empirical data? Is there not an essential level of abstraction in such physics where the math IS the physical model?
Click to expand...

This is getting really philosophical. What is the difference between
mathematics and physics, and where is the bordeline? Since
the models are still presumed to say something about the real
world, although not necessaily about what it is really like since
they are just models, I guess we still have to say it is physics.
However, it may be that the physical models add very little to
the mathematical theories, perhaps. I guess this is a question
for the physicists to chew on. SY or anyone listening?
 
#71
Christer --

Then we are coming from the same background. I am also a computer guy, but in university I took the introductory classes in theoretical math (Analysis, Algebra, Topolgy etc.) as well as theoretical computer science (Crypto, Comutation, Complexity etc).

My statement to V was a brief restatement of my first post, which you basically identified as the set-theory version of the Peano Axioms. So, I'm assuming you don't actually take issue with the "infinite number of definitions problem" because you can obviously describe it more compactly than that. You probably know lambda calculus. Same thing.

I'm not sure what you mean by syntax versus semantics here. I'm not trying to define the numerals (syntax?) but the concept of the natural numbers (semantics?). Is that what you mean?

I guess I was a little bit confused about your infinity+1 construction. It seems that it should be equal to infinity, just as it is in mathetmatics, but perhaps it is distinguishable somehow in your programming system? It seems odd, but then again I don't know any logic programming. Now I'm kind of curious about this, come to think of it.

What I'm having trouble reconciling is the existence of this thread given your knowledge. Why entertain that Goldbach's Conjecture stuff when you know that you're right?

-Won
 
#72
Won said:
Then we are coming from the same background. I am also a computer guy, but in university I took the introductory classes in theoretical math (Analysis, Algebra, Topolgy etc.) as well as theoretical computer science (Crypto, Comutation, Complexity etc).
Click to expand...

Yes, I suspected you might be a CS too, or maybe you even
said so. One problem I have realized over the years is that
computer science is such a broad area that nobody knows
enough of the basics. Or rather, everybody seems to have
a different mix of courses, depending on which university they
studied at. Some focus more on semantics for programming
languages, others on logic, others on algorithms and complexity
etc. etc. Many don't focus much on theory at all. Besides, when
i got my degree, we still didn't have many real theoretical CS
courses, so I took kind of the experimental first version of some
of them. I have later tried to patch up my knowledge in those
areas where I have needed to know more for my research.


My statement to V was a brief restatement of my first post, which you basically identified as the set-theory version of the Peano Axioms. So, I'm assuming you don't actually take issue with the "infinite number of definitions problem" because you can obviously describe it more compactly than that. You probably know lambda calculus. Same thing.
Click to expand...


Yes, since you seemed a bit picky about what I said I though
I should be a bit picky about what you said. 🙂
My CS background had quite a bit of focus on logic, so that
sometimes makes me overly sensitive to certain claims,
formulations etc. The issue was whether we can actually
define all natural numbers, or just construct them. I suppose
many mathematicians wouldn't bother about this, but logicians
would.

Actually, Lambda calculus is one of those things I have avoided
since I have not been very interested in the semantics of
programming languages. I guess I know the "tourist version" of
it though, since I have programmed a lot in Lisp and also
tought Lisp courses with some emphasis on its semantics.


I'm not sure what you mean by syntax versus semantics here. I'm not trying to define the numerals (syntax?) but the concept of the natural numbers (semantics?). Is that what you mean?
Click to expand...

Well, if you haven't taken any logic course, you probably won't
know the difference, since mathematics usually doesn't bother
about it. In fact, one logician I spoke to once even was bold
enough to claim that most mathematicians don't really know
what they are doing since they know too little logic and can't
distinguish syntax from semantics. He added, though, that
it usually doesn't matter for most branches of mathematics,
they get the rigth result anyway.


I guess I was a little bit confused about your infinity+1 construction. It seems that it should be equal to infinity, just as it is in mathetmatics, but perhaps it is distinguishable somehow in your programming system? It seems odd, but then again I don't know any logic programming. Now I'm kind of curious about this, come to think of it.
Click to expand...

I have only bothered about this once, when I took a course in
logic programming long ago, and had to learn this. I don't quite
remember why it was needed or how it was used. Howver,
this is a generalization of the natural numbers called ordinals,
and which is straightforward to do in the set-theoretic model
of the natural numbers. I don't remember the exact details, but
I think it worked in the way I described earlier. Anyway, this
makes it possible to "count above infinity". We cannot count up
to infinity, of course, but by using ordinals, we can start counting
at infinity and enumerate ordinals like N+1, N+2,.... Similarly,
we can start counting at "twice the infinity", but I'd better not
extrapolate too much from memory to try to remember the
definition of that. Anyway, it's called ordinals if you wish to
look it up somewhere.


What I'm having trouble reconciling is the existence of this thread given your knowledge. Why entertain that Goldbach's Conjecture stuff when you know that you're right?
Click to expand...

Maybe you never saw the thread where this started, and I think
I have already forgotten how we got there in that thread.
What happened was that Millwood said something like "it is
not proven whether 1+1". He didn't mention anything about
the context of the Goldbach conjecture, however, so there
was no specific context in which to interpret it. From my logic
point of view, it isn't possible to prove or disprove something
like "1+1", since it is a term, not a logic formula. unfortunatly
the thread was closed before I could ask millwood for a
clarification and it wasn't possible to email him, which is why
I started this thread. It was quickly resolved, when he explained
he meant it as a shorthand notation relating to the Goldback
conjecture. However, then a lot of other people started to
discuss various related/non-related issues, which is why we
are where we are now. 🙂
 
#76
Re: Re: Re: MRT, etc.

planet10 said:
real analysis is a subset of analysis, another is complex analysis.
Click to expand...

Interesting, since several of you have used the term analysis,
I am starting to wonder if I have been mislead about the
english terminology. The first and very big math course we
usually take at university covers the standard stuff on real
and complex numbers, including derivation, integrals, sums
and series, often also all this for multi-variable functions.
We call this course "analys" in swedish but in the english
descriptions of curses it is usually translated to "calculus",
hence I always call this calculus in english. Is this a mistranslation,
perhaps, and the correct term should be "analysis" also in
english??


eStatic said:
And Christer, thanks for tolerating my digressions.
Click to expand...

No problem. My original question was sorted out in the first
few posts, so the floor is open. 🙂

I find your questions deep and interesting, although I suspect
some of them have no simple or satisfactory answer.
 
#77
eStatic, I think a better example than Hilbert spaces might be non-Euclidean geometries. Certainly, working before Poincare, Lorentz, and Einstein, Riemann had no clue that his replacement fifth postulate would end up with something useful that actually describes real space; it was just an interesting exercise in pure mathematics, to make a completely consistent system using a different set of postulates than Euclid.

Is it not true that once one moves away from the realm where Newtonian mechanics applies, that is, into the realm of un-intuitive physics, models of existence are primarily or entirely mathematical? I.E. many of the crucial concepts require, in order to be persuasive and complete, mathematical expressions consistent with their defining empirical data? Is there not an essential level of abstraction in such physics where the math IS the physical model?
Click to expand...

In a sense, yes. But one develops a different set of intuitions with experience. We come into physics with a rather Machian view of the world, where everything is intuited according to our slow, macroscopic existence. And when you study QM, for example, all that stuff about particle exchange, tunneling, cooperative phenomena, non-local phenomena, acausality, time reversal and the like are totally non-intuitive- at least they were for me. And they are indeed dealt with in a very abstract way, of necessity. I mean, try to find some pictures in Dirac's "Quantum Mechanics"!

But the workers in those fields develop a different set of intuitions, though these are not easily accessible to rank amateurs like me (how does a string theorist "imagine" a nine dimensional function?).

I still think, in my heart of hearts, like Mach, the Universe as a collection of gears, poles, and pivots, with my eyes open that if I want to understand a QM problem, I've got to use a mathematical formalism. But I understand that smarter, more experienced guys can bring intuition, a different and better sort of intuition, to bear on these problems. It's just tough when you have to explain to someone why a photon from a flashlight in a spaceship will have the same measured speed to the astronaut in the ship and the observer on the ground...
 
#78
Just an anecdote on the issue of comprehending difficult
concepts.

It is said that a young PhD student in math once approached
one of the big authorities in math (it might have been Hardy,
but memory fails me) and asked if he had some good advice on
how to understand the more difficult concepts in mathematics.
The older collegue said, young man, in mathematics you don't
understand things, you just get used to them.

I suppose there is actually a bit of truth to this, especially
since mathematics is axiomatic and not necessarily intuitive.
Maybe the same holds for some modern branches of physics?
 
#80
Won said:
something can be both deterministic and unpredictable
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An interesting example has been used to argue that a sort of free will can exist in a non-quantum, deterministic world: you cannot predict what will happen to the system you are in since you also have to predict yourself predicting yourself, ad infinitum.
Of course I don't see this as free will, since not being able to know your destiny does not equate with not having one. And of course adding QM does not give free will either, since the nondeterminism is just randomness. Reducing free will to non-mental processes takes the magic away, and it's not really free will any more. The alternative is panpsychism (mind is a fundamental, irreducible part of the universe), and most philosophers rightly find that notion unacceptable. Free will is subjective only.
 
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