[reply]Frank you are rather missing the point, we are talking about Maths here, not Physics. I plucked an equation from Newton mechanics. You could replace F=ma with Q=CV (charge = capacitance x p.d) for an electromagnetism equation or A = LW (area of a square = length x width) for a geometry equation or anything (A = BC).

I chose the Newtonian equation because it's probably recognisable to most triathletes as its a pretty important equation in their sport (even if it doesn't work when you're cycling at relativistic speeds).

How do you calculate C^2 if you have no photons? (i.e E = 0 and m = 0).[/reply]

I understand what you were trying to do.

Here is my problem with some of the math absolutists. Math starts with a few "givens" and everything follows from there. And, when things don't quite fit in, then, it seems, when it is convenient, they "define" something to be so, like 0^0=1 but in other instances they prefer to say something is undefined rather than defining it as something. But, change one or more of the givens (add a couple of dimensions or change the surface from flat to a saddle shape or a donut) and you have a completely different math with completely different results to the same question (is the shortest distance between two points really a straight line).

Most of this math is way beyond my ability (or desire) to comprehend but I enjoy the conjecture and trying to stave off the degeneration by keeping the old brain active, even though some of you think these posts are proof the oldtimers disease has already set in.[quote]

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Frank,
An original Ironman and the Inventor of PowerCranks

You said "toss." Heh, heh, heh.

The missing "-1" is obviously the key to disproving this "proof". Your explanation is much more eloquent than mine would be. (I wonder why?)

:)

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king of the road says you move too slow
KING OF THE ROAD SAYS YOU MOVE TOO SLOW

[reply]Tough to discuss relativity with someone that doesn't understand 0/0...[/reply]

No it is tough to discuss relativity with someone who DOES understand 0/0, or, at least, claims to.

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Frank,
An original Ironman and the Inventor of PowerCranks

Yeah, that's right Frank, I claim to know... my bad for getting into this.

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"Yeah, no one likes a smartass, but we all like stars" - Thom Yorke


smartasscoach.tri-oeiras.com

[reply]But, change one or more of the givens (add a couple of dimensions or change the surface from flat to a saddle shape or a donut) and you have a completely different math with completely different results to the same question[/reply]

This might make CCF's or Francois' head explode but...

Read Gregory Chaitin's new book "Meta Math! The Quest for Omega". It was published last month, and IS NOT new age gobbly-goop.

Here is the snippet from the Time magazine article about him last month:

THE OMEGA MAN

Over the past few decades, Gregory Chaitin, a mathematician at IBM's T.J. Watson Research Center in Yorktown Heights, N.Y., has been uncovering the distressing reality that much of higher math may be riddled with unprovable truths--that it's really a collection of random facts that are true for no particular reason. And rather than deducing those facts from simple principles, "I'm making the suggestion that mathematics is done more like physics in that you come about things experimentally," he says. "This will still be controversial when I'm dead. It's a major change in how you do mathematics."

Chaitin's idea centers on a number he calls omega, which he discovered in 1975 and which is much too complicated to explain here. (Chaitin's book Meta Math! The Quest for Omega, out this month, should help make omega clear.) Suffice it to say that the concept broadens two major discoveries of 20th century math: Gödel's incompleteness theorem, which says there will always be unprovable statements in any system of math, and Turing's halting problem, which says it's impossible to predict in advance whether a particular computer calculation can ever be finished.

Sounds like a nonevent in the real world, but it may not be. Cryptographers assume that their mathematically based encryption schemes are unbreakable. Oops. "If any of these people wake up at night and worry," says Chaitin, "I'm giving them theoretical justification." --By Michael D. Lemonick. Reported by Matt Smith/New York

The only survivors to a plane crash in the desert are an Engineer, a Physicist and a Mathematician. The only food they have is lieing about the wreckage in tin cans, unfortunately no one can find a can opener. The three thought together for a while until the Engineer said "Why don't we throw the cans against that rock over there and bash them until they break?". They tried but all they could do was dent the cans. Then the Physicist said "we don't need to do anything, just leave the cans to be heated in the sun, then cool down at night and so on until the can breaks because of the thermal expansion", but they all knew it would take too long, and probably wouldn't work anyway ..... then the Mathematician added his opinion!!! He stood up, held the can up in front of him and boldly proclaimed "Assume can to be open!".

That was the punchline by the way. However, my point is this: Engineers and Physicists have their so called 'givens' already laid out for them by the real world. A Mathematician on the other hand can choose his own and create whole new worlds (on paper) that could be n-dimensional and where the shortest distance from A to B is a curve and so on. I wouldn't make any distinction between a 'given' and a definition, they are one and the same as long as you work them through from the start.

And that is why I am currently in my 4th year studying Mathematical Physics at Nottingham University in the UK.

As for staving off the degeneration by keeping the old brain active I am only too happy to help :), I am only 21 so I guess I'm lucky (for now).

(btw defining 0^0 to be 1 is essentially done to save paper, so you don't have to write out an extra bit on the end of each equation explaining what happens for x=0, which is usually not a very interesting result anyway)

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http://mclean.errl.org.uk

[reply]As for staving off the degeneration by keeping the old brain active I am only too happy to help :), I am only 21 so I guess I'm lucky (for now). [/reply]

When to start worrying is when you think you are getting smarter and beginning to understand all this stuff. A sure sign old timers disease is starting. You probably have a few years to go.

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Frank,
An original Ironman and the Inventor of PowerCranks

Nah, my head's not exploding (yet). Although you have peaked my interest, I'd like to know what this "number" he claimed to discover is. In logic omega is most commonly used to denote the cardinality of the natural numbers. If this guy is claiming to have "discovered" this in 1975, then he's full of it. But I'm guessing it's something different, otherwise it would never have been published.

One of the beauties, in my mind, of mathematics is that there is room for all types of arguments such as this. If the logicians want to argue that our mathematical systems are unstable and "riddled with unprovable truths", then they can. If you don't want to do this, then you can shoot over to a completely different field of math, and have at some of the problems there!

I find higher logic (and higher CS, which is really just math) quite interesting, but it doesn't float my boat. So I stay with algebra and, in particular, group theory. Good stuff.

Actually, the existence of proofs that, in our current system of mathematical logic, are "correct", is the main reason I left astrophysics and started doing math. I was tired of physics profs getting stuff wrong and saying that they had "proved" things, when in reality all they had done was make a few observations.

-C

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Any run that doesn't include pooping in someone's front yard is a win.

[reply]Nah, my head's not exploding (yet). Although you have peaked my interest, I'd like to know what this "number" he claimed to discover is. In logic omega is most commonly used to denote the cardinality of the natural numbers. If this guy is claiming to have "discovered" this in 1975, then he's full of it. But I'm guessing it's something different, otherwise it would never have been published.

One of the beauties, in my mind, of mathematics is that there is room for all types of arguments such as this. If the logicians want to argue that our mathematical systems are unstable and "riddled with unprovable truths", then they can. If you don't want to do this, then you can shoot over to a completely different field of math, and have at some of the problems there!

I find higher logic (and higher CS, which is really just math) quite interesting, but it doesn't float my boat. So I stay with algebra and, in particular, group theory. Good stuff.

Actually, the existence of proofs that, in our current system of mathematical logic, are "correct", is the main reason I left astrophysics and started doing math. I was tired of physics profs getting stuff wrong and saying that they had "proved" things, when in reality all they had done was make a few observations.

-C[/reply]

I find your response interesting. One of the amazing things about humans is our ability to adapt to the needs of the task at hand. both you and Francois work in a field where proof is taken seriously and must be rigourously applied yet both of you were able to determine with enough satisfaction that PC's were of benefit to you, even though the usual level of proof that you require in your work was lacking. So, you were able to adapt the level of proof you required to make a decision that was situation specific.

We do this all the time from requiring very rigorous proofs in some situations to absolutely forgiving total lack of proof when it comes to religeous belief (it is called faith for a reason, those who try to prove the existence of God I think don't get the concept). We vary in our ability to make these "proof" distinctions however which is what makes for somewhat heated discussions.

Anyhow, the discussions regarding these issues can be fun, even to those who, like me, are clueless to some of the esoterics of these disciplines.

Frank

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Frank,
An original Ironman and the Inventor of PowerCranks

[reply]
We do this all the time from requiring very rigorous proofs in some situations to absolutely forgiving total lack of proof when it comes to religeous belief (it is called faith for a reason, those who try to prove the existence of God I think don't get the concept). We vary in our ability to make these "proof" distinctions however which is what makes for somewhat heated discussions.
[/reply]

The heated discussions is what confuses me about science vs. religion sometimes because if the two areas would communicate better, maybe the discussions wouldn't be so heated. Physics and mathematics both rely on a great deal of faith. It's a different flavor of faith than is generally used in religion, but it's still the same beast.

Physics is accurate in its predictions up to the precision of the uncertainty principle, and mathematics is accurate up to the precision of a numerical programs ability to complete a summation for a set of numbers approaching zero or infinity. Science is unavoidable empirical (which is what Chaitin's book focuses on, he simply "discovered" a number, omega, to quantify it).

For CCF:
What I love about astrophysics is that the field is ever evolving and we have avoided the minefield of being "correct" like high energy physicists or solid state physicists. Ever talk with a researcher in quantum computing or particle physics!? God they are infuriating.

And this is why I'm an engineer, not a mathemetician. I sat through 4 semesters of calculus with my eyes glazing over, but give me an application, and voila! It makes sense!

no no no no no....mathematics is not accurate up to whatever...numerical analysis is and it is a very small part of mathematics.
When you do mathematics, you have a set of axioms and you prove results. And these results are TRUE in this framework. Not more not less.
Mathematics are certainly not empirical.

[reply]no no no no no....mathematics is not accurate up to whatever...numerical analysis is and it is a very small part of mathematics.
When you do mathematics, you have a set of axioms and you prove results. And these results are TRUE in this framework. Not more not less.
Mathematics are certainly not empirical.[/reply]

Exactly what I was about to say.
-C

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Any run that doesn't include pooping in someone's front yard is a win.

Yeah, hard core physicists can be pretty weird. When I say I wanted to get out of astrophysics, what I really mean is that I wanted to get out of physics. Astronomy is great.

You're coming to IMFL, right? I can picture it now: You, Francois, and I, dead tired after the race, trying to figure out the mysteries of the universe. THAT will be an interesting conversation.

-C

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Any run that doesn't include pooping in someone's front yard is a win.

Oh man, I knew that might drive you nuts. I guess we've touched on the dicotomy in science, the Aristotelian view of things vs. the Platonic view. I'll agree to disagree with you on the empirical view of math. But I'll point to one example that bothers me a great deal:
- the E.M. field equation in physics (in tensor form) can be used in conjunction with the continuity equation to derive all of Maxwell's equations, and adding a gravity term also allows derivation of Einstein's equations. This is a pretty powerful little tool that fell out of math theory and seems to be confirmation of the "true" nature of mathematics... but, it's only as useful as the continuity equation, which can only be proven empirically. That's strange. Lower order math is very tidy and impenetrably consistent, but higher order math (like tensor residue systems, or abstract geometry, or quantum topology) aren't quite so... obvious.

Take a look at Chaitin's book, but be forewarned, it will probably leave you in disgust.

it doesn't matter if experience confirms or not...if you have a set of axioms and prove a result, it's true. period. the semantics of it in a given field has nothing to do with the truth of the result.

I disagree. I think that the problem is that physicists are trying to use certain types of math to solve physical problems that said types of math can't solve. So the physicists make some leaps, which are fine in the world of physics, but sometimes leave the math cloudy, or worse, incorrect. When viewed as math in it's own right (math for math's sake), it's always tidy.

-C

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Any run that doesn't include pooping in someone's front yard is a win.