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### Re: Philosophy of Mathematics

what would the interpretation of the square root of a probability be? something in particular, something we can imagine in the "real" world? or just useful as math in intermediate calculations?
Interesting!

In the frequentist world, the square root would relate event density of an area to intra-event distances in that space, no? bearish
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### Re: Philosophy of Mathematics

If q is the probability of the joint occurrence of two independent and equally likely events, then the probability of each one of them occurring is given by the square root of q. So it can be made to mean something. Collector
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### Re: Philosophy of Mathematics

Thanks this is interesting..probably the square root of the two answers are interesting I mean. Cuchulainn
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### Re: Philosophy of Mathematics

what would the interpretation of the square root of a probability be? something in particular, something we can imagine in the "real" world? or just useful as math in intermediate calculations?
I suppose quantum amplitude (it is complex-valued), its absolute valued is equal to a probability.  So take sqrt.
What it means in maths will be different than a more intuitive physics motivation? It might be some kind of wave equation, whatever..

Maybe the Quantum Computer expert here can help. qubits,what?

// Are you still positive about negative probability? It will probably break down here (sqrt(p <0) is complex?)
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R. van Gulik Cuchulainn
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### Re: Philosophy of Mathematics

If q is the probability of the joint occurrence of two independent and equally likely events, then the probability of each one of them occurring is given by the square root of q. So it can be made to mean something.
Is the square root of a probability also a probability in general?
As in real/complex analysis for given $x$ find (unique or otherwise $y$ such that $y^2= x$. In this case it it a complex number.

$\sqrt{x}$ is not a fundamental quantity. It is a kind of formalism. It's the same process as defining  the integers ${...,-2,-1,0,1,2,...}$ as ordered pairs of  the natural numbers ${...,0,1,2,...}$ BTW this was the analogy what Feynman alluded to in his note on negative probability although he did not do the axioms AFAIK.

Very parsimonous. The trick is to get new results with as few axioms as possible and these results should build on previous results and not by pulling them out of a magician's hat. Number systems are built incrementally. Should be same with probabilities IMO.
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R. van Gulik Posts: 23951
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### Re: Philosophy of Mathematics

Generalizing bearish's insight suggests that if the product of probabilities may be a probability, then the roots of a probability may be a probability.

The deeper issue is that a probability is not just an ordinary number that can be freely manipulated with no implications on validity or meaning. A probability is always attached to another logical construct -- an event and an event space usually in the context of a larger system. It's the logic of the events and the system that determine how respective probabilities might be mathematically manipulated (e.g., products and roots) whilst retaining their status as probabilities. Cuchulainn
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### Re: Philosophy of Mathematics

Generalizing bearish's insight suggests that if the product of probabilities may be a probability, then the roots of a probability may be a probability.

This is unfounded. One swallow makes a summer not. The word 'may' is never used in mathematics.

IMO probability is a special kind of measure; 'event' is looser. Anything you prove should (?) be done through Lebesgue.
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R. van Gulik Posts: 23951
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### Re: Philosophy of Mathematics

Generalizing bearish's insight suggests that if the product of probabilities may be a probability, then the roots of a probability may be a probability.

This is unfounded. One swallow makes a summer not. The word 'may' is never used in mathematics.

IMO probability is a special kind of measure; 'event' is looser. Anything you prove should (?) be done through Lebesgue.
The phrase 'IMO' is never used in mathematics either. I was merely saying there exists at least one subset of probabilities for which the square roots are also probabilities. That subset is a subset of the set of probabilities that are products of other probabilities (bearish's example). The existence of other such subsets is not known to me. Cuchulainn
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### Re: Philosophy of Mathematics

Yeah, remove IMO with regard to probability. It is many many years since I did that course (my prof was a PhD student of W. Feller at Princeton).

IMO -> AFAIR.
Last edited by Cuchulainn on November 1st, 2017, 3:30 pm, edited 2 times in total.
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R. van Gulik Collector
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### Re: Philosophy of Mathematics

"the word 'may be' is never used in mathematics."

because it was replaced with probabilities? so the question boils down to if we can have $\sqrt{\mbox{May-be}\times \mbox{May-be}}$

It seems like it even works for negative quasi probabilities.

What now Cuch, Mayday ? Cuchulainn
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### Re: Philosophy of Mathematics

The situation is serious, but not hopeless. I wouldn't pull the plug just yet.
Maybe work out a couple use cases on bearish' suggestion.
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R. van Gulik Collector
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### Re: Philosophy of Mathematics

Why Square Roots of Probabilities?

"Square roots of probabilities appear in several contexts, which suggests that they are somehow more fundamental than probabilities. "
a bit strange reasoning I think.

" the roles that square roots (or roots) of probabilities play in various applications  have an unclear and less well-understood foundational significance."

"promise to resolve some of these mysteries surrounding square roots of probability"

square root of probabilities used in quant finance ? (except naturally in Planck scale HFT) Cuchulainn
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### Re: Philosophy of Mathematics

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R. van Gulik Cuchulainn
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### Re: Philosophy of Mathematics

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R. van Gulik outrun
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### Re: Philosophy of Mathematics

or in this case, the same as two exactly the same cases.  