About Lawrence Krauss vs Hamza Tzortzis debate
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20-12-2015, 06:56 PM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 04:39 AM)aguelmame Wrote:  Dear ZoraPrime,
There are actually some mathematicians, although not many, trying to redo mathematics without the recourse to real numbers (see for instance Norman Wildberger, Youtube channel Here. Actually, he calls people using real numbers "believers" !).
If you read french, you can also have a look at an interesting paper about the necessity of ininity in mathematics Here .

I find it enriching to think about what Pr. Eugene Wigner called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I think we can learn about the universe by asking theses questions (Paul Dirac used to say that if a physical law was not mathematically "elegant", it is likely to be incorrect).

Physicists use real numbers and I don't see that changing. Every now and then, a particular quantity in a particular circumstance will be discrete (e.g. energy levels in bound states in quantum mechanics), but by far and large, quantities are real unless noted otherwise. Sure, it's technically possible that every physical quantity is fundamentally discrete; but we have no reason to suspect so. I just don't see the point in speculating.

And while i haven't read Wigner's work, mathematics is understandably reasonable in natural science. You won't be applying mathematics to biology, for example, because such systems are too complicated to be modeled effectively by mathematics. And so much of physics now are using computers to solve equations numerically because elementary solutions often times do not exist. The fact that the universe obeys mathematics just reflects the fact that the universe evolves in a (more or less) predictable way and we can quantify obsersables.

but if this excerpt is anything to go by, Wigner's argument is full of crap. yes, pi shows up everywhere in integrals. I may give you the integral sqrt(1-x^2) from -1 to 1 and tell you that that's pi/2. You might say "but how can pi possibly appear?" except, such an integral has a lot to do with circles. In the case of the Gaussian, exponential functions have a very intimate connection with trigonometric functions. Whether or not these connections are "unexpected" is entirely dependent your point of view. Does the fact exp(ix)=cos(x)+isin(x) seem unexpected? It did to me at first, so much so that I really focused on what defines an exponential; but now I understand how the exponential is defined (it's the thing when differentiates returns itself times a constant), Euler's Formula seems obvious in retrospect. Does the distribution property seem unexpected, i.e. that 2(x+y)=2x+2y? to seem people it does, since they'd think 2*(x+y)=2x+y; moreover, the distributive property is usually an axiom so nothing fundamental can explain why they're wrong. Except, when you consider where that axiomatic system came from, i.e. by counting objects in the real world, in which case the distributive property seems obvious because it models the real-world.

And that's the problem with Wigner's argument. he's arguing that these unexpected connections invalidate any understanding why have on physics because mathematics just happens to be coincidental. But mathematics ia a language plus logic; the rules of logic we study in mathematics are chosen precisely because they correspond to the logic we see in the real-world. The usefulness of mathematics largely reflect the fact that our universe is logical. We may not understand why our universe is logical (i.e. evolves in a predicable way) at the most fundamental level, but we see that everyday we can say it's provisionally true. This isn't even considering the fact that biology cannot use mathematics because, as logical as biology is, the systems are far too complex. but since wigner was a theoretical physicist who probably kept to problems that he could solve mathematically (he'd be out of the job otherwise), he probably was too myopic to realize that other disciplines don't attack everything with differential equations first, questions later.
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20-12-2015, 07:04 PM (This post was last modified: 20-12-2015 07:10 PM by Chas.)
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 04:39 AM)aguelmame Wrote:  
(20-12-2015 02:28 AM)ZoraPrime Wrote:  Eh.

all of the mathematics used to formula the laws of physics as we know them today necessarily involve infinity at least implicitly. Transcendental numbers, such as pi, are completely unavoidable--the permeability of free space, for example, is defined exactly in terms of pi. However, all these numbers are infinite inasmuch as you need to use a mathematical limit as something goes to infinity. "Limits as something goes to infinity" can be defined in a way such that don't make reference to infinity.

Are physical quantities necessarily finite? Who knows. Gravitational singularities are expected from Einstein's equations, and black holes may be such that example. I'm not sure if these have been observed; don't pay that much attention to astro. In general, yes we expect observable physical quantities to be finite to be observable, but that's just because we haven't observed an infinite observable.

In terms of Krauss's comment, I see it as a mater of semantics more than anything. Are we gonna include mathematical machinery as "real"? That's really the heart of the question, and I don't find hat an interesting question to answer, because by the time you get to answering you really wouldn't have learned one iota about the universe. Mathematics needs infinity (particularly limits thereeof) to make sense including mathematics physicists use; but we generally don't expect observables to be infinite. That's all you're going to come back to, to be honest.

Dear ZoraPrime,
There are actually some mathematicians, although not many, trying to redo mathematics without the recourse to real numbers (see for instance Norman Wildberger, Youtube channel Here. Actually, he calls people using real numbers "believers" !).
If you read french, you can also have a look at an interesting paper about the necessity of ininity in mathematics Here .

I find it enriching to think about what Pr. Eugene Wigner called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I think we can learn about the universe by asking theses questions (Paul Dirac used to say that if a physical law was not mathematically "elegant", it is likely to be incorrect).

Can you summarize what the problem with real numbers is supposed to be? Consider

You will note that 4 is a real number as is 1/2.

The set of natural numbers is a proper subset of the set of integers,
the set of integers is a proper subset of the set of rational numbers,
and the set of rational numbers is a proper subset of the set of real numbers.

Skepticism is not a position; it is an approach to claims.
Science is not a subject, but a method.
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20-12-2015, 09:13 PM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 06:56 PM)ZoraPrime Wrote:  
(20-12-2015 04:39 AM)aguelmame Wrote:  Dear ZoraPrime,
There are actually some mathematicians, although not many, trying to redo mathematics without the recourse to real numbers (see for instance Norman Wildberger, Youtube channel Here. Actually, he calls people using real numbers "believers" !).
If you read french, you can also have a look at an interesting paper about the necessity of ininity in mathematics Here .

I find it enriching to think about what Pr. Eugene Wigner called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I think we can learn about the universe by asking theses questions (Paul Dirac used to say that if a physical law was not mathematically "elegant", it is likely to be incorrect).

Physicists use real numbers and I don't see that changing. Every now and then, a particular quantity in a particular circumstance will be discrete (e.g. energy levels in bound states in quantum mechanics), but by far and large, quantities are real unless noted otherwise. Sure, it's technically possible that every physical quantity is fundamentally discrete; but we have no reason to suspect so. I just don't see the point in speculating.

And while i haven't read Wigner's work, mathematics is understandably reasonable in natural science. You won't be applying mathematics to biology, for example, because such systems are too complicated to be modeled effectively by mathematics. And so much of physics now are using computers to solve equations numerically because elementary solutions often times do not exist. The fact that the universe obeys mathematics just reflects the fact that the universe evolves in a (more or less) predictable way and we can quantify obsersables.

but if this excerpt is anything to go by, Wigner's argument is full of crap. yes, pi shows up everywhere in integrals. I may give you the integral sqrt(1-x^2) from -1 to 1 and tell you that that's pi/2. You might say "but how can pi possibly appear?" except, such an integral has a lot to do with circles. In the case of the Gaussian, exponential functions have a very intimate connection with trigonometric functions. Whether or not these connections are "unexpected" is entirely dependent your point of view. Does the fact exp(ix)=cos(x)+isin(x) seem unexpected? It did to me at first, so much so that I really focused on what defines an exponential; but now I understand how the exponential is defined (it's the thing when differentiates returns itself times a constant), Euler's Formula seems obvious in retrospect. Does the distribution property seem unexpected, i.e. that 2(x+y)=2x+2y? to seem people it does, since they'd think 2*(x+y)=2x+y; moreover, the distributive property is usually an axiom so nothing fundamental can explain why they're wrong. Except, when you consider where that axiomatic system came from, i.e. by counting objects in the real world, in which case the distributive property seems obvious because it models the real-world.

And that's the problem with Wigner's argument. he's arguing that these unexpected connections invalidate any understanding why have on physics because mathematics just happens to be coincidental. But mathematics ia a language plus logic; the rules of logic we study in mathematics are chosen precisely because they correspond to the logic we see in the real-world. The usefulness of mathematics largely reflect the fact that our universe is logical. We may not understand why our universe is logical (i.e. evolves in a predicable way) at the most fundamental level, but we see that everyday we can say it's provisionally true. This isn't even considering the fact that biology cannot use mathematics because, as logical as biology is, the systems are far too complex. but since wigner was a theoretical physicist who probably kept to problems that he could solve mathematically (he'd be out of the job otherwise), he probably was too myopic to realize that other disciplines don't attack everything with differential equations first, questions later.

Not so- Biomathematics isn't all that new a field.

http://biomath.usu.edu/htm/faq/careers
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21-12-2015, 12:13 AM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 09:13 PM)Fireball Wrote:  Not so- Biomathematics isn't all that new a field.

http://biomath.usu.edu/htm/faq/careers

I meant in the sense every inch of bio isn't covered in mathematics a la physics. Mathematics isn't absent in biology; it's just nowhere near as pervasive as math is in physics.
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21-12-2015, 05:42 AM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(21-12-2015 12:13 AM)ZoraPrime Wrote:  
(20-12-2015 09:13 PM)Fireball Wrote:  Not so- Biomathematics isn't all that new a field.

http://biomath.usu.edu/htm/faq/careers

I meant in the sense every inch of bio isn't covered in mathematics a la physics. Mathematics isn't absent in biology; it's just nowhere near as pervasive as math is in physics.

But it's still pretty pervasive. Anything dealing in genetics or populations will heavily involve statistics among other mathematics. It's easy enough to explain and understand concepts like genetic drift or inheritance without the use of math, but math is integral in building up and interpreting the evidence those concepts are built upon.

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21-12-2015, 11:51 AM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(21-12-2015 05:42 AM)EvolutionKills Wrote:  
(21-12-2015 12:13 AM)ZoraPrime Wrote:  I meant in the sense every inch of bio isn't covered in mathematics a la physics. Mathematics isn't absent in biology; it's just nowhere near as pervasive as math is in physics.

But it's still pretty pervasive. Anything dealing in genetics or populations will heavily involve statistics among other mathematics. It's easy enough to explain and understand concepts like genetic drift or inheritance without the use of math, but math is integral in building up and interpreting the evidence those concepts are built upon.

Biology (even without genetics) uses quite a bit of mathematics, such as the χ² analysis (tests results for prediction accuracy) and the fractal analysis of healthy biospheres, in terms of distribution of a species throughout an area (such as trees of varying stages of growth). Genetics, except for the actual process of sequencing DNA, is almost entirely mathematical, and as EK pointed out, is essential to understanding not only why evolution happens, but how exactly it is so-- not only for probabilities (of inheriting a given trait, such as the Punnet Square a la Mendel) but also for how much proximity of genes on a chromosome influences their mutual expression, how proteins fold, and how much of a population a given set of resources can sustain... and so on, and so on.

Sure, it's not as pervasive as in physics, where practically everything is math, but it basically works like this: math tell us how physics works, physics tells us how chemistry works, and chemistry tells us how biology works. Essentially, a physicist need only concentrate on math (and perhaps know a bit of chemistry in terms of Atomic Theory, if they're a particle physicist, but need not know biology at all), a chemist must know math and physics in order to calculate why the atoms bind into molecules the way they do, and a biologist must know math, physics, and (at least organic) chemistry in order to do their job. But in truth, it's all one big field, and all are integral to one another.

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21-12-2015, 02:01 PM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 07:04 PM)Chas Wrote:  Can you summarize what the problem with real numbers is supposed to be? Consider

You will note that 4 is a real number as is 1/2.

The set of natural numbers is a proper subset of the set of integers,
the set of integers is a proper subset of the set of rational numbers,
and the set of rational numbers is a proper subset of the set of real numbers.

I would like first to state an important point : I don't have any problem with real numbers or infinity. Moreover, I love Cantor work !

When I talk about real numbers, I am implicitely refeering to irrational numbers (elements of R\Q).

According to some mathematicians, the problem with irrational numbers is that in principle, no actual physical measure can output an irrational number with certainty.
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21-12-2015, 03:16 PM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 06:56 PM)ZoraPrime Wrote:  but if this excerpt is anything to go by, Wigner's argument is full of crap. yes, pi shows up everywhere in integrals. I may give you the integral sqrt(1-x^2) from -1 to 1 and tell you that that's pi/2. You might say "but how can pi possibly appear?" except, such an integral has a lot to do with circles. In the case of the Gaussian, exponential functions have a very intimate connection with trigonometric functions. Whether or not these connections are "unexpected" is entirely dependent your point of view. Does the fact exp(ix)=cos(x)+isin(x) seem unexpected? It did to me at first, so much so that I really focused on what defines an exponential; but now I understand how the exponential is defined (it's the thing when differentiates returns itself times a constant), Euler's Formula seems obvious in retrospect. Does the distribution property seem unexpected, i.e. that 2(x+y)=2x+2y? to seem people it does, since they'd think 2*(x+y)=2x+y; moreover, the distributive property is usually an axiom so nothing fundamental can explain why they're wrong. Except, when you consider where that axiomatic system came from, i.e. by counting objects in the real world, in which case the distributive property seems obvious because it models the real-world.

I have few remarks about your mathematical assertions :
1- There are a lot of identities where Pi is involved with no link in any way to circles or trigonometry (see for instance the work of Ramanujan). I always find it fascinating to see Pi emerging when unexpeced.
2-Exponential : There are actually many "defining" properties of the exponential. One that generalize best to manifolds is that of "one parameter sub-group", precisely, a group homomorphism from (R,+) to (G,.), where G is a general Lie group. The classical exponentiel can be defined as a group homomorphism between (R,+) and (R*,.) (where '.' is multiplication).
3- For the case of natural numbers, distributivity is a consequence of commutativity and associativity of addition : 2*(x+y) = (x+y)+(x+y) = x+y+y+x=x+2*y+x = 2*x + 2*y. It is then natural to require this for subequent extensions of natural numbers (Z,Q,R,C,...)
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22-12-2015, 10:10 AM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(21-12-2015 02:01 PM)aguelmame Wrote:  According to some mathematicians, the problem with irrational numbers is that in principle, no actual physical measure can output an irrational number with certainty.

I can't imagine a mathematician seeing that as a problem, only engineers and the like who actually measure things. Consider

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22-12-2015, 02:51 PM
RE: About Lawrence Krauss vs Hamza Tzortzis debate
(20-12-2015 06:56 PM)ZoraPrime Wrote:  Physicists use real numbers and I don't see that changing. Every now and then, a particular quantity in a particular circumstance will be discrete (e.g. energy levels in bound states in quantum mechanics), but by far and large, quantities are real unless noted otherwise. Sure, it's technically possible that every physical quantity is fundamentally discrete; but we have no reason to suspect so. I just don't see the point in speculating.

Better to say that although it is actually the premise of quantum mechanics that every observable is quantized at a sufficiently fine scale, that does not imply that all quantities are - probability, for one, is not. Qubit phase is not only real but complex.

I'd agree even more strongly with your last line there. In that if the OP even has a point in mind, they haven't gotten there yet.

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