## Sunday, 23 June 2013

### Strange Beliefs about Abstract Objects

Occasionally, the strange belief that mathematical objects are physical objects is advocated. I am truly baffled when I hear such beliefs. Here are the some questions:
Is the number $0$ a physical object?
Is the number $2^{2^{2^{2^{2^{2^{2^{2}}}}}}}$ a physical object?
Is the wellorder $(\omega, <)$ a physical object?
Is the topological space $\mathbb{R}^4$ a physical object?
Is the Lie group $SU(3)$ a physical object?
Is the rank $V_{\omega + 57}$ a physical object?
For example, what is the mass of $(\omega, <)$? Can you find it somewhere, perhaps at Tesco's?

#### 5 comments:

1. I would believe this only if this was advocated by abstract objects.

Seriously, where has this been advocated?

2. Hi AJ JA

Ultrafinitists make this claim. E.g.,

http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Ultrafinitism.html

They insist, e.g., that because a physical computer, say, is finite, there aren't numbers beyond what it represents. But this philosophical conclusion requires the further assumption that numbers *are* physical objects. This assumption is the one that is not justified.

Cheers,

Jeff

3. Hi again AJ JA,

It is advocated by E.B. Davies (see E.B. Davies, 2005: "Some Remarks on the Foundations of Quantum Theory", Brit. J. Phil. Sci. 56, p. 530.)
See

http://m-phi.blogspot.co.uk/2011/04/2-become-1.html

Cheers,

Jeff

4. Hi AJ JA,

Another example - the physicality of mathematical objects is advocated by Doron Zeilberger:

"(ii) the traditional real line is a meaningless concept. Instead the real REAL ‘line’, is neither real, nor a line. It is a discrete necklace! In other words R = hZp, where p is a huge and unknowable (but fixed!) prime number, and h is a tiny, but not infinitesimal , ‘mesh size’. Hence even the potential infinity is a meaningless concept."

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

Zeilberger's philosophical view here simply assumes that mathematical objects (such as $\mathbb{R}$) must be *physical*. Otherwise, the relevant concept is, he says, "meaningless".

But the reasonable view is that $\mathbb{R}$ is an abstract object, and its connection, if any, to the physical world is something to be investigated empirically.

Cheers,

Jeff

5. Thanks, Jeff!

Cheers,
A