That’s the definition of *i*: the square root of -1. That’s why *i* doesn’t correspond to a regular number.

The surreals is a good example I think, right now anyway. And I assume the reals are “real”, in the sense we are trying to discuss. But I can see at least three problems that could, but won’t, make this a very long discussion:

1) What if the surreals suddenly did become almost indispensable in physics (or at least some set of which the reals formed a proper subset)? Surely it would be better that any division of mathematical beasts into the “real” and the others should be independent of time. Are the quaternions now real, but not before? The Mandelbrot set? The tangent bundle to spacetime? (Horrible name-dropper I am!)

2) Surely also many people would regard as ‘realer than the reals in general’ the computable real numbers, a very small set despite the fact that the rest of the reals are essentially undescribable, even un-namable, individually.

3) Is the set of all real numbers real? Or is it just some (all) individual ones? Is it the reals from a set theory where the continuum hypothesis holds, or one where it doesn’t?

I’d be inclined to think that the whole exercise of having some distinction like this to be a mistake.

]]>I would guess that by Francis definition, surreals would be an example of math that isn’t meaningful for physics. Maybe “mathy”, but not “real-y”.

For myself, I think Deutsch has a useful, so meaningful, definition of realism, constrained reaction on constrained action. (“If you hit a stone, it hits back.”) In that sense all mechanics tests for realism (and so a robust, interdependent, and measurable with uncertainty, system) from the get go. (Action-reaction, observation-observables.) They would have to, or observation isn’t consistent with that it works.

While that doesn’t get to what is real, except that observations (experiments) are themselves examples of systems attributed with the same realism as the systems we observe, it says that mathematics is games we play in order to make observations and else for fun.

Why anyone would try to attribute a characteristic of “real” for mathematical objects is curious and an effort apriori likely to fail. Any such reification would make them magical objects I think, as they are not under observation. Maybe we should invent a category of “magical numbers” for such? ;-)

]]>Can you give an example of something in math…which is NOT real in a meaningful sense?

]]>Thanks also to commenter Peter for replying to my question.

]]>My statement “it cannot be the same line as the real axis” should have read ‘it cannot be parallel to the real axis’ more generally. Taking it parallel but not equal to the real axis would also give a contradiction. It’s good exercises to verify what I have left out, but anyone wanting to see those things I’d be glad to provide (and will continue to read this thread of this blog).

]]>Firstly “…the idea of a square root being a rotation…” is a perfect illustration of a reader being mislead by the blogger’s error. It is not a rotation any more than absolutely every complex valued function of a complex variable is a rotation, and that would be a misleading use of the word ‘rotation’, one which does not agree with what a mathematician calls a rotation. (Sorry to just repeat what I said earlier, but the blogger’s response to that earlier correction was unsatisfactory.)

Secondly, the reader asks a good question: “Is (the artifice …) a convention or is there a deeper mathematical logic to it? “. The blogger did not answer it. The answer is the following. You want to have beforehand a nice classical geometrical plane with rotations, orthogonality etc., and to take the real numbers as the horizontal axis. You want to take all the real multiples of ‘i’ as going along a line. All that is good and necessary. The question is: which line should the imaginary axis be; in particular, should it be perpendicular to the real axis? The answer is that

1) first and pretty obviously it cannot be the same line as the real axis. That seems trivially obvious, and would definitely lead to an easy logical contradiction if you took the same line; but more to the point,

2) it can in fact be any other line. So, yes, the fact that the square root function has a nice geometric description (though not a rotation), including describing the angle of the answer, is an artifice of choosing the imaginary axis perpendicular.

On the other hand, why not choose that geometric description of the axis, since it leads to such a simple geometric description of many other important things, including that of square roots.

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