At *Closer to Truth,* the mathematical physicist Roger Penrose explains to Robert Lawrence Kuhn how he understands the relationship between mathematics, the mind, and the physical world:

Penrose:Then in the physical world we have these conscious beings and these conscious beings are part — a very small part — of a physical world… So it’s a very small part of the physical world which seems to have direct relationship to consciousness. And I regard this consciousness as having a different kind of existence but it springs from that very tiny part of the physical world. (3:07)But in the world of conscious experience, we also have understanding and we have understanding of mathematics. That again is a very tiny part of mentality. But nevertheless that tiny part of the mentality … in a sense encompasses or at least has the potential to encompass the top world, which is the mathematical world. And I sort of draw this as a kind of paradox because it’s a small part of each world which seems to encompass the totality of the next one. And it’s deliberately drawn as paradoxical just to emphasize the strangeness of this thing. (3:50)

Penrose attempts a minimalist position when defending the reality of both mathematics and the mind in a world where many believe that only the physical exists.

There are no triangles in the real world only in the mind.

However, all these abstract non real world concepts are incredibly useful in the world where they do not exist. What would we do without them?

That is the amazing thing.

We’ve been here before—you keep forgetting the Bermuda Triangle. Tell all those lost seamen triangles don’t exist. And then, also, what about the thousands of “YIELD” signs virtually everywhere……?

Folks, of course, mathematically perfect triangles are abstract entities that will be manifested in any possible world as part of its necessary, embedded logic of structure and quantity. The two 7″ speed squares [one, Al, the other fibreglass plastic] next to me as I type, illustrate how pervasive triangle properties are . . . especially the protractor side. At the same time, they show that approximation and controlled tolerance are key engineering phenomena, their points are rounded, not razor sharp, their markings are builder-coarse not set square fine, but not even my TD set squares and tee square are abstractly exact, never mind that their corners are close enough to needle sharp to require care. Extending, optical precision of the zoom lens on a camera also next to me, are also less than mathematically exact but are good enough for purpose. Then, magic step, the logic and structure of tolerance, subject matter of one of my uni textbooks, “errors of observation and their treatment,” is itself a highly ordered mathematical discipline. And of course, as my Victorinox lockback shows, even a razor sharp edge is not a mathematical point. But, engineering approximation is itself pervaded by a deeper level of Mathematics. KF

PS, the Penrose triangle [is it the same Penrose?] shows how our optical system expresses computational processing and how its context-sensitive response can lead to optical illusions. Each part seems to be part of a 2-d structure but they are incongruous and no such structure can be built.

F/N: Apparently, it is both Penroses, Father [a psychiatrist] and Son [a Mathematician].

You know, you could have just said Swiss Army knife so we didn’t have to look up Victorinox lockback …. 🙂

CD, Wenger, then quite active, and the lockback was a specific model. I also have a champ and a couple of others. KF