# FOM: arithmetic, geometry, natural science, formal systems, ...

Harvey Friedman friedman at math.ohio-state.edu
Thu Oct 15 18:10:06 EDT 1998

```Tennant 5:30PM 10/15/98 writes:

>Steve Simpson writes:
>_________________________
>I want to suggest that the correct logic for natural
>science is classical rather than intuitionistic.  Identity,
>non-contradiction, excluded middle: these are bedrock for me, because
>reality is real, independent of our understanding of it.
>Intuitionistic logic may be useful for other purposes, e.g. as a
>description of *our understanding of reality at a given point in
>time*, because if we are not now in a position to confirm or deny X,
>then neither X nor not X is now known.  But natural science is the
>study of what's real, not the study of *our current understanding of*
>what's real.  That's why I focus on classical rather than
>intuitionistic logic.
>__________________________
>
>Wouldn't this require an argument to the effect that only classical (but not
>intuitionistic) logic can handle certain inferences that are needed by the
>natural scientist?

No. For example, it is commonly believed that it is easiest to use
classical analysis for natural science than intuitionistic analysis,
including such things as the intermediate value theorem, attainment of
maxima, etcetera, and to be able to use such things as "if x is not <= 0
then x > 0," etcetera. These are not intuitionistically provable under
normal setups.

Any kind of absolute necessity of using classical logic is another matter
entirely. Claiming that it is far easier to use classical logic is much
weaker than it being absolutely necessary.

>But it seems that there is a good argument to the contrary. If you're a
>classicist, you will not mind if anyone re-writes all your laws of nature
>that have the form "All F's are G's" as ones in the form "It is not the case
>that there is an F that is not a G". (Indeed, Popper once recommended doing
>exactly this, in order to appreciate what a law of nature of so-called
>"universal" form is "really saying". What it's really saying, according to
>Popper, is that we won't find a counterexample: an F that is not a G.)

I am not convinced that this is what one is really saying. I think it one
of many interesting views on the matter, which lead to significant f.o.m.
And a classicist would in fact mind this restatement since it will prove
awkward in actually writing papers and doing research.

>In other words, you (the classicist) won't mind if the language of science
>does without the universal quantifier. Let's call the resulting language L.

"Won't mind" is likely to be false as indicated above.

>Informal Claim: All that Logic is needed for in the course of doing natural
>science is to produce such *refutations* as there may be of falsifiable
>theories about the external world. (Using the logic, we extract predictions
>from the hypotheses; then we compare these with the results of observation.
>If prediction and observation clash, absurdity results.)

The status and meaning of your "Informal Claim" is not clear when
mathematical analysis (rigorous analysis) is involved.

>Metatheorem: Any classical proof of absurdity from assumptions X in L can be
>matched by (indeed: effectively transformed into) a proof in *intuitionistic
>relevant logic* of absurdity from (some subset of) X.

It is not clear what matched by means here if mathematical analysis
(rigorous analysis) is involved. Also, in contexts such as arithmetic where
it is clearer what this means, what does "can be" mean? For example, I
happen to know that your version of relevant logic is a fragment of cut
free logic. And mathematics cannot be done *by people* in a cut free
system.

>Informal Conclusion: There's no compelling reason to use classical logic
>to develop natural science, no matter how much of a `realist' you happen
>to be in your philosophical outlook.  Whatever you do (refutationally)
>with classical logic, you can do with intuitionistic relevant logic.

There's no compelling reason not to use classical logic to develop natural
science, no matter how much of an 'intuitionist" you happen to be in your
philosophical outlook.

>Strictly classical math is therefore
>a ladder that can be kicked away at the end of the day. Or, perhaps more
>aptly, a crutch that we can throw away.

Given recent work on brain transplants,
http://www.the-times.co.uk/news/pages/tim/98/10/15/timfgnusa01006.html?1124027,
we can or will be able to communicate with the outside world without legs,
arms, or even a voice. So legs, arms, voices are all crutches we can throw
away.

>Steve, what's your reaction to this "Amahl" phenomenon?  As a tough-minded
>fellow, shouldn't you too throw away your crutches (i.e., the law of
>excluded middle)?

As an active fellow, shouldn't you too use facilitating methods (i.e., the
law of excluded middle)?

```