Mathematics/Logic

[Doyle_Mashern]

I can see how it might be tempting to think of right angled triangles as things which exist in some kind of platonic realm, and perhaps in the mind of God, but even numbers themselves can be defined in a way which has nothing to do with metaphysics.

I must confess to being somewhat of a closet platonist. However, I’m not suggesting a stark realm of the forms concept. Numbers not only can, but must be defined in a way which is independent of metaphysics. Designing a bridge needs to be a pragmatic exercise.

As a Christian though (Big Bang type), I believe that all physical things ultimately come from the mind of the Creator. There is an inherent beauty in mathematics. The beauty of the sunflower is a function of the fibonacci sequence found in the pattern of its seedheads. Aesthetics and mathematics are inseparable.

The real beauty of mathematics comes from the fact that it is the language of creation. People who have the intellectual acuity to understand complex mathematical forms and relationships (you but certainly not me) are able to see a beauty that not all can see. However, the rest of us can appreciate and be deeply moved by things like art, music and architecture because the mathematical relationships at their core resonate with our mathematical hard-wiring to light up our brains and our souls. This goes well beyond the simple pragmatism of arithmatic.

I don’t care that I can’t sing like a Bocelli or don’t have the money of a Gates. But I do wish I could understand complex mathematics.

 

[JonW]

Is anyone here enthusiastic about Mathematical Logic?

God Bless

Jon

 

[mathematician]

Is anyone here enthusiastic about Mathematical Logic?

God Bless

Jon

Not especially, although I prefer it as the philosophical basis of mathematics to the alternatives.

 

[JonW]

So I’ve noticed. I can’t say that I’m with you on that though.

God Bless

Jon

 

[mathematician]

So I’ve noticed. I can’t say that I’m with you on that though.

God Bless

Jon

That is what comes of having it drummed into you - mathematics is about nothing if not rigorous proof.

 

[JonW]

Sorry, I’m quite tired now. Could you flesh that out a bit? Thanks

jon

 

[mathematician]

Sorry, I’m quite tired now. Could you flesh that out a bit? Thanks

jon

During the first year of a maths degree, at least in the UK, you repeatedly find yourself being asked to prove seemingly self evident things. Until you begin to think like a mathematician your immediate reaction is, “What do you mean prove it? It’s obvious!” Unfortunately, “It’s obvious” doesn’t count as a proof. You have got to be able to give a tightly argued mathematical account of why it’s “obvious”; justifying every single assertion you make along the way. Getting you to think in that ultra-rigorous fashion takes up just about the first year of a three year degree course. Once they are satisfied that you are beginning to think like a mathematician in the making then they let you move onto more interesting things in the second and third year, unless you decide that mathematical logic is really your thing; in which case you will spend the rest of your life splitting epsilon deltas.

 

[IvanKaramozov]

Is anyone here enthusiastic about Mathematical Logic?

God Bless

Jon

Somewhat, I’m a Freshman so I’m still at relational predicate logic, but soon I think I should be able to get into more complex subjects

 

[East_and_West]

Does anyone here know what to do with the apparent contradictions or paradoxes that can be found in set theory? The seem to destroy mathematical and deductive logic in general. I haven’t really had to deal with these issues since college but as I prepare to go back to school to study philosophy, especially Aquinas, I would like to know how to resolve these problems found in math.

 

[mathematician]

Does anyone here know what to do with the apparent contradictions or paradoxes that can be found in set theory? The seem to destroy mathematical and deductive logic in general. I haven’t really had to deal with these issues since college but as I prepare to go back to school to study philosophy, especially Aquinas, I would like to know how to resolve these problems found in math.

I assume that you are referring to things like Russell’s Paradox. They are not likely to destroy mathematics; the paradoxes just mean that you have to be more careful about how you define sets, and the allowable things you can do with them. In other words you need an axiomatic system upon which to base set theory. A number of such systems have been drawn up, the best known of which is ZFC (Zermelo-Fraenkel, with the C being appended when the Axiom of Choice was included as an axiom). Another such system is BNG (Bernays-Neumann-Godel).

 

[East_and_West]

I assume that you are referring to things like Russell’s Paradox. They are not likely to destroy mathematics; the paradoxes just mean that you have to be more careful about how you define sets, and the allowable things you can do with them. In other words you need an axiomatic system upon which to base set theory. A number of such systems have been drawn up, the best known of which is ZFC (Zermelo-Fraenkel, with the C being appended when the Axiom of Choice was included as an axiom). Another such system is BNG (Bernays-Neumann-Godel).

Such this does not bring into question the validity of deductive logic?

 

[mathematician]

Such this does not bring into question the validity of deductive logic?

No it doesn’t bring into question the validity of deductive logic. We would be in big big trouble if it did. The mathematicians could pack up and go home for a start.

 

[IvanKaramozov]

Such this does not bring into question the validity of deductive logic?

validity is a property of deductive logic, to say that deductive logic was not valid would be somewhat incoherant woulden’t it?

Metamathematics ans metalogic as well as type theory, I beleive, clear up most of the antinomies

 

[East_and_West]

validity is a property of deductive logic, to say that deductive logic was not valid would be somewhat incoherant woulden’t it?

Metamathematics ans metalogic as well as type theory, I beleive, clear up most of the antinomies

Cool. I am just checking as I am not an expert in set theory.

 

[JonW]

Hi mathematician,

Here in the states we have a formal class in which you learn how to do proofs. It utilizes some set theory, number theory, logic, combinatorics, etc., and yes, many of the first proofs were so simple as to seem to defy proof, etc. It’s been my experience that, in many classes and even more so in many texts on any mathematical subject, elementary proofs play an important part in aqcuainting the reader with the subject in the beginning.

Still, I don’t see that this shows math logic as the “philosophical basis” of mathematics. Also, what do epsilon-deltas have to do with math logic? Perhaps you’re thinking of Topology? Anyway, my field is Model Theory, a part of logic. The field has had far reaching applications to many fields in higher math. The most recent example was Hrushovski’s proof of the Mordell-Lang Conjecture for function fields. I like to apply model theory to algebra and algebraic geometry.

God Bless

Jon

 

[mathematician]

Still, I don’t see that this shows math logic as the “philosophical basis” of mathematics.

I am thinking more of the mindset it induces, rather than a formal link. The mindset being why I go for mathematical logic as the philosophical basis, rather than a mindset which has mathematical objects floating around in a platonic ether somewhere, or a mindset which has mathematics almost as an offshoot of physics.

Also, what do epsilon-deltas have to do with math logic? Perhaps you’re thinking of Topology?

I was thinking of real variable analysis.

Myself, I prefer my sets to be infinite dimensional vector spaces, with a metric defined upon them.

 

[JonW]

Myself, I prefer my sets to be infinite dimensional vector spaces, with a metric defined upon them.

As in Differential Geometry?

 

[mathematician]

As in Differential Geometry?

As in functional analysis.

 

[I_like_Mike]

Hi mathematician,

Here in the states we have a formal class in which you learn how to do proofs. It utilizes some set theory, number theory, logic, combinatorics, etc., and yes, many of the first proofs were so simple as to seem to defy proof, etc. It’s been my experience that, in many classes and even more so in many texts on any mathematical subject, elementary proofs play an important part in aqcuainting the reader with the subject in the beginning.

Still, I don’t see that this shows math logic as the “philosophical basis” of mathematics. Also, what do epsilon-deltas have to do with math logic? Perhaps you’re thinking of Topology? Anyway, my field is Model Theory, a part of logic. The field has had far reaching applications to many fields in higher math. The most recent example was Hrushovski’s proof of the Mordell-Lang Conjecture for function fields. I like to apply model theory to algebra and algebraic geometry.

God Bless

Jon

Is there a problem with topology as that was the area that my model designs originated from?

Essentially in an energy field two galaxies drifting apart would be reducing the gravity of the void between them leading to a lower energy limit. That is where I ran into mathematicians who would alternate between yes there is vacuum energy and then no space can not deform into a whirlpool between galaxies but then yes the Penrose lowering of energy could form the basis of a theoretical worm hole.

Finally there could be no linked structure through space despite the deep gravity lensing images showing a web across the universe. But what gets me is that if the dark energy is real as science insists it is then all of space is linked by it and expected to expand uniformly because it must be applied across a common centre of 4D expansion.

That was just the big stuff then the quantum issues where there is ‘supposedly’ an inner quantum medium but in order to do anything normal protons have to be used. Incidentally protons have a diameter the same order of magnitude to the smallest Planck scale as to the approximate size of the universe. So when I asked about a central boundary and protons are the most stable things we have, laughter.

Well I left them with the thought that the simpler creatures of the world are not bound by logic and if for some reason an experiment is going to go tragically wrong … all of the world’s animal depending on their sensitivity will scream shortly before the change is upon us. The reply was the deletion of the thread.

I think it is because I also said that gravity probe B will not point where it is supposed to … and it didn’t. Sure one of the four gyros pointed roughly but not accurately at the star, two of the other gyros pointed sideways and the fourth was pointing backwards. Censorship or what?

 

[I_like_Mike]

I am not saying I am right, just that it was an idea. Putting the boundary central solved the complex counter intuitive mathematics required to describe a micro structure with its biggest problem aside from disqualifying magnetism is that it is essentially ‘back to front’.

Then quantum could act as a reflection of reality down to the smallest level without having to invert. On the large scale it posed a problem of other mirror symmetries such as seeing a reflection of our own world every 2150 years due to velocity in a large structure and at every pole change due to acceleration.

Anyway enough of that. If one does not speak the language of mathematics it is clear that even in the interest of discussing an idea, any idea is not possible. For ordinary folk with a bit of an interest in the world regular mathematicians on the science sites seem to go out of their way to be unhelpful, or am I just mistaking the application of correct skeptical thinking?