This page is an exercise in organizing my own thoughts -- I am certainly not claiming any novel insight!
What is the relationship between logic/mathematics and the physical universe?
Religion -- for instance, Christianity insisting on God as creator of
the physical universe -- implicitly places such questions within some broader world-view; I won't discuss this
religious view any further.
To the question
there are two basic possible answers ("yes" and "no"!), and let me elaborate briefly.
Do the rules of logic (or mathematics, though logic seems more basic) exist and make sense independent of the nature of the physical universe?
seems sensible, and indeed such questions are often discussed in connection with
of the universe to "life as we know it"
(For longer discussions see
Giberson - Collins).
At an opposite extreme, does it make sense to discuss the question
(1) how would the universe be different if the numerical value of the fine-structure constant were slightly different?
One can certainly invent and analyze some mathematical equations for hypothetical different forces;
but what is the conceptual status of such an exercise?
Scientists and mathematicians tend to implicitly take what I will call
- (2) what can we say about a hypothetical universe in which the
four fundamental forces of physics
did not exist but were replaced by some other forces?
Implicit in this view is that (1) and (2) have the same status as "rational
analysis of a hypothetical scenario",
it's just that (2) is more speculative.
An alternative "common sense" view of these two questions is that
(1) is as close to real science as anything not experimentally testable can be;
whereas (2) is closer to writing a fantasy novel. By analogy, it is fun
speculating on what a lion would say if it could talk, but we do not usually
consider such speculation as "rational analysis".
So common sense asks: if talking lions cannot be discussed rationally, how can
changing all the rules of the physical universe be discussed rationally?
- the absolutist view:
the rules of logic and mathematics do exist
and make sense independent of the nature of the physical universe.
What is an alternative to the absolutist view?
Well, our unconscious mental processes evolved in the physical universe, so to
be useful they must be broadly consistent with how the physical universe operates.
Subsequently, our conscious notion of logic must have been built upon these unconscious
This prompts what I will call
- the intertwined view:
our view of logic
is inextricably mixed with our perception of physical reality.
This is the view I adopt, though it can certainly be criticized for sidestepping
rather than resolving the issue.
The Unreasonable Effectiveness Of Mathematics
It's just a small variation on the previous topic to ask
This is often regarded as a basic philosophical question, and discussed under the phrase deriving from
a 1960 article
The Unreasonable Effectiveness Of Mathematics In The Natural Sciences
My only comment is that from the absolutist view it certainly is a mystery: why indeed
should a physical universe conform to mathematics, if mathematics is an entirely different category of thing?
From the intertwined view it's no mystery.
Just as the fact that eyes allow us to see useful things is no mystery (eyesight evolved because it was useful),
so we emphasize mathematics as part of our intellectual world-view because it has proved useful.
A popular discussion of these topics is in Chapter 9 of
Is God a Mathematician?.
Why can mathematics be used to predict certain fundamental aspects of the physical world?