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Cuchulainn
Posts: 63378
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### Re: Philosophy of Mathematics

So, negative probabilities as per Feynman are clearly kosher. Any others are a terrible heresy and the speaker is likely a witch (gender neutral). I suggest the ordeal by water test to find out when in doubt.
Now, Cardinal Fang, read the charges
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262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

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Cuchulainn
Posts: 63378
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### Re: Philosophy of Mathematics

Reality
• An engineer thinks that equations are an approximation to reality.
• A physicist thinks reality is an approximation to equations.
• A mathematician doesn't care.

So, a physicist encounters funny numbers when working equations (which are correct, of course) so just give the number a new name.

Why can't negative probabilities be defined axiomatically? Feynman mentions negative numbers in his note in an offhand way. It is really this.

https://en.wikipedia.org/wiki/Integer

What is a np, really? Is it really an equivalence class of pairs of normal probabilities? Whatever..

Basically, NP := a number system with operational rules.
My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

http://www.datasimfinancial.com
http://www.datasim.nl

katastrofa
Posts: 9707
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Philosophy of Mathematics

From what I understand Feynman reasons that since we can have negative numbers (-5 apples), why not have assign negative values of probabilities? He takes the total probability formula (Eq. 1) and assigns negative values to the conditional probabilities (small p's). Conditioning simply means that one is counting the events of interest only in the sub-set of all possible events, which meets this condition - let's call it A after Feynman. However, if this conditional probability is not positive, there are no such events in the subset (they do not occur in situation A). Since we know that they can occur anyway (because Feynman is talking about them), they must be somewhere outside of this subset - in some another subset, i.e. under another condition, call it B. Those subsets add up to the full set of possible events and they don't overlap - otherwise Feynman couldn't have used the total probability equation. So where is Nemo? (Nemo is dead - overfishing.) In which condition should I add those events in the total probability formula - the one under which they can't occur or the one under which they can occur? And if I also have conditions C, D, ..., should I treat events as positive probabilities in respective subsets or rather as negative probabilities in the complementary set? Equivalently, which subsets are the pre-images of the Borel sets on which I defined the probability in the real space? I wonder how Feynman magic clarified that, befores he moved on to using ...

Another explanation of Feynman's idea is even more daunting: he seems to assume that if probability is a type of a measure and there exist signed measures which can be have negative values, then probability can be negative too. Measure can indeed be extended to negative numbers, to complex numbers, vectors, Borel alpacas or whatever one wishes as long as it's additive and consensual. Probability is indeed a type of a measure. Howeverm, not all measures are probabilities. This paper is so wrong that it's dificil to argue.

On a side note, there are situations in which negative probabilities crop up in models of natural phenomena - it's only a signal that there is more to them than a simple theoretical model or our understanding assume. Most physical systems obey the assumptions of the Kolmogorovian probability theory  (Kolomogorov isn't saint either - he cut corners of mathematical rigour when introducing his calculus, but it's a different story). However, sometimes we cannot fulfil these assumptions (e.g. the statistical stabilisation or reproducibility assumptions) and it leads to anomalies - death of reality, Bell inequalities, ghost particles, etc. But they witness barely the limit of our mathematical map of nature rather than represent real physical problems.
Negative energies (somehow mentioned on this occassion) of bound states simply mean that you need to perform some work/provide energy to the system to change its state to a one with positive energy. For example an electron in a hydrogen atom has much lower energy compared to a free electron with respect to its centre of mass with a free proton. If you want the latter as zero on the energy scale and then starts to bring the two particles closer to each other, their energy will go down the negative scales, because they will be bound by the Coulomb interaction, which attracts them - and in the darkness binds them (-: Who's having LOTR maraton for X-mas? I'm watching The Witcher this year and I think it's great.

katastrofa
Posts: 9707
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Philosophy of Mathematics

Still, Hussite Trilogy is Sapkowski's opus magnum: https://en.m.wikipedia.org/wiki/Hussite_Trilogy
Tw, you might like it. History instead of Slavic mythology, mysticism and boobs.

tw
Posts: 1176
Joined: May 10th, 2002, 3:30 pm

### Re: Philosophy of Mathematics

Still, Hussite Trilogy is Sapkowski's opus magnum: https://en.m.wikipedia.org/wiki/Hussite_Trilogy
Tw, you might like it. History instead of Slavic mythology, mysticism and boobs.
Thanks for the recommendation! Sounds intriguing.  (although mythology etc. have their charms...)
I have so say, my reading list is growing quicker than my bandwidth at the moment.
Alan also recommended "The genius of birds" on another thread. When the Illuminati agree in public , it must be a fantastic

bearish
Posts: 5994
Joined: February 3rd, 2011, 2:19 pm

### Re: Philosophy of Mathematics

Maybe I'm not doing this right, but I'm still looking for the boobs.

trackstar
Posts: 27636
Joined: August 28th, 2008, 1:53 pm

### Re: Philosophy of Mathematics

Maybe I'm not doing this right, but I'm still looking for the boobs.
With The Genius of Birds in mind, here is a Boobie for you. : )

And if that is not quite what you had in mind, may I suggest that we return to the Philosophy of Mathematics.

I don't think the Principia Mathematica has been mentioned yet (not Newton, but Whitehead and Russell.)

Principia Mathematica - Stanford Encyclopedia of Philosophy

Also, an interesting discussion of the PM and how it influenced the development of Wolfram Mathematica.

100 Years Since... Stephen Wolfram Blog

Cuchulainn
Posts: 63378
Joined: July 16th, 2004, 7:38 am
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### Re: Philosophy of Mathematics

Maybe I'm not doing this right, but I'm still looking for the boobs.
You're on the wrong page!

Page 3, prego
My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

http://www.datasimfinancial.com
http://www.datasim.nl

Cuchulainn
Posts: 63378
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### Re: Philosophy of Mathematics

Gil Strang on teaching Linear Algebra

My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

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http://www.datasim.nl

Paul
Posts: 10915
Joined: July 20th, 2001, 3:28 pm

### Re: Philosophy of Mathematics

Gil Strang on teaching Linear Algebra

Excellent! I agree 100%. The only thing I’d suggest is that he uses more politically incorrect jokes and analogies in his lectures.

Cuchulainn
Posts: 63378
Joined: July 16th, 2004, 7:38 am
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### Re: Philosophy of Mathematics

Gil Strang on teaching Linear Algebra

Excellent! I agree 100%. The only thing I’d suggest is that he uses more politically incorrect jokes and analogies in his lectures.
He's too nice a man for that. BTW he is my academic grandfather.

GS wrote the first mathematical book on Finite Element  in 1973. On the inset, he wrote the joke "Cherchez la fem".
Last edited by Cuchulainn on January 7th, 2020, 7:54 pm, edited 2 times in total.
My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

http://www.datasimfinancial.com
http://www.datasim.nl

Paul
Posts: 10915
Joined: July 20th, 2001, 3:28 pm

### Re: Philosophy of Mathematics

The nicest people I know are politically incorrect! WYSIWYG.

Cuchulainn
Posts: 63378
Joined: July 16th, 2004, 7:38 am
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### Re: Philosophy of Mathematics

The nicest people I know are politically incorrect! WYSIWYG.
Jeremy Corbyn is non PC but still you don't like him?
My C++ Boost code gives
262537412640768743.999999999999250072597198185688879353856337336990862707537410378210647910118607313

http://www.datasimfinancial.com
http://www.datasim.nl

Paul
Posts: 10915
Joined: July 20th, 2001, 3:28 pm

### Re: Philosophy of Mathematics

The nicest people I know are politically incorrect! WYSIWYG.
Jeremy Corbyn is non PC but still you don't like him?
A) You cannot be serious!

B) Sigh.

DrBen
Posts: 500
Joined: February 8th, 2003, 1:24 pm

### Re: Philosophy of Mathematics

All was not lost; at the time Schwartz, Sobolev and others took the delta function by the scruff of the neck and generalize to the theory of Distributions, without which there would be no Finite Element Method.

The concrete example again..

So, instead of huffing and puffing with Fourier and integral, just go for the Cauchy distribution?
This work shifted the entire philosophy when studying (linear) PDEs (i.e. significant portion of work of mathematical community). Prior to the Theory of Distributions people would consider existence and uniqueness of solutions within a given set of functions for a specific or class of PDEs. Once you view a function as a distribution and have the associated interpretation for (linear) PDEs and the analogue framing of the Fourier Transform, you are able to make the following step. Apply Fourier Transform to any (linear) PDE will yield a solution (which is a distribution), that is within the set of distributions all linear PDEs defined on then will yield a solution which is also a distribution. Therefore, distributions are to linear PDEs what complex numbers are the polynomials. This single step shifts the discussion from arguments concern existent of solutions to studying the properties of solutions.

[Comment: I put linear PDE everywhere because I do not know situation with non-linear PDEs, but be interested if could extend above points (or version of) to any non-linear PDEs. Saying this, as far as I know most non-linear PDEs studied come of description of physical processes and so existent as such is always assumed since people case about the properties of solutions, which correspond to properties of physical process and explanation for these processes.]

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