Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, May 09, 2007

Mach Speed

In the first paragraph of Principles of the Theory of Heat [1], Mach states, "The collection of these instances of the physical behaviour of a body, which are connected with the mark of our sensations of heat - the collection of reactions - is termed its thermal state, or state with respect to heat."

Mach was the first great modern relationalist. In his writing, the observable properties of bodies are considered to define physical states, and this pragmatism is taken far more seriously than the concept of universal (theoretical) law. Later, Mach discusses causality: "Where we assign a cause, we only express a relation of connection, an existing of fact; that is to say, we describe. ... It is far better to regard the conceptual determinative elements of a fact as dependent upon one another in exactly the same sense as the mathematician, for example the geometer, does."

This is very much in the spirit of relational set theory, the axioms of which are properly formulated in topos theory. In particular, the axiom of comprehension states that there exists a set (contained in a chosen universal set) containing all elements $x$ such that $x \in \phi$, where $\phi$ is a property that may be stated logically. Recall that it was Gray who first considered extending this axiom to category theory, whereby he found himself developing the Gray tensor product for bicategories.

[1] in English, ed. B. McGuinness (Reidel 1986)


Blogger L. Riofrio said...

I first encountered his name through the mach numbers of supersonic aircraft. Mach's ideas of cosmology are a continued inspiration.

May 09, 2007 4:16 PM  
Blogger CarlBrannen said...

This cartoon precisely expresses what I think of Mach's principle: appropriate cartoon.

May 10, 2007 8:19 AM  
Blogger Kea said...

Sigh, Carl. Why does everybody think that Mach only wrote about spinning buckets?

May 10, 2007 8:36 AM  
Blogger CarlBrannen said...

Well he was very influential, so if the authorities had stressed to the younger generation the fact of Mach's repeated and vehement denying of the existence of atoms, it would have made all physics look silly, and the foundations of both quantum mechanics and relativity, which use Mach's principles, appear shaky.

The funny thing is that these same authorities had no problem going on and on about what Lord Kelvin thought about physics or the sun and thermodynamics, or later, what Einstein thought about QM. As in most of life, who you know (and how badly you've pissed them off) matters a lot.

That the cartoon showed up in today's newspaper was clearly some sort of sign from somewhere. Which reminds me. Have you seen the latest ESP stuff from physics Nobelist Brian Josephson?

The really amazing thing is that I find the results very natural. I've been expecting this for many decades. The reason is that the only available mechanisms to connect body and soul are subtle enough to have evaded science.

This means they have to rely on the "butterfly effect". That is, the soul makes very small influences that are amplified into large changes in the body's actions. And that means that they have to have look ahead into the future. And that means that the soul operates over more than just a single moment of time.

Otherwise there could be no free will.

May 10, 2007 9:31 AM  
Blogger Kea said...

I don't mind if we don't have free will (which is never properly defined, even by J. Conway et al) if all we can know is the semblance of it. And in my opinion, Mach was not influential enough in physics, although in philosophy and other domains he certainly was. His anti-atomism was about mechanistic thinking, not about quantum atoms. To quote a footnote of Mach's: I have, therefore, no objection to make when Boltzmann praises the advantages of the atomic theory, above all other conceptions, for the physicist. The investigator is not only permitted, but he is expected, to employ all means that can assist him. I should be misunderstood, if a bias for the assumption of a continuous plenum were attributed to me....All that I oppose is the permanent adherence to arbitrary accessories of the facts.

May 10, 2007 10:04 AM  

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