From: *Bruno Marchal* <[email protected] <mailto:[email protected]>>
But now, let us move forward. Stop saying “realism or platonism”, in
pour metaphysical context this lead to misunderstanding.
Assuming classical arithmetic = arithmetical realism.
It is becoming clear that we have very different understandings of what
is meant by arithmetical (or mathematical) realism. I gathered a few
statements about realism from your recent post -- included here:
"Realism = classical. Realism means that I use the axiom (A v ~A)."
"All scientific theories use arithmetical realism, but you are still
using it in a philosophical/metaphysical sense, when it means simply
that we accept the excluded middle principle in arithmetic."
"Of course, given what I mean by arithmetical realism (which I thought I
already told you), this would mean that you reject the use of (A v ~A)
in arithmetic"
My understanding of 'realism' comes from the idea of scientific realism.
This can have a number of nuanced interpretations, but the basic idea of
scientific realism is a cluster of views about the nature of scientific
theories and theorizing. A common core might be the following:
(1) The aim of scientific inquiry is to produce theories that provide
description of the world that are literally true.
(2) Theories in the 'mature sciences' are usually approximately true,
and the entities postulated by those theories usually exist.
One of the most popular arguments for scientific realism is the
so-called 'miracle argument'. Following Putnam, scientific realism is
capable of explaining why a predictively successful theory is
predictively successful, whereas the success of a theory would be
miraculous if scientific realism were not true.
Stathis Psillos adds a metaphysical component: the world has a definite
mind-independent structure; and a semantic component: scientific
theories are truth-conditioned descriptions of their intended domain, so
the theoretical terms in theories have factual reference -- the
unobservable entities they posit populate the world -- form the
'furniture' of reality.
The Oxford Dictionary of Philosophy, in the section on the philosophy of
mathematics, gives the following definitions:
"There are two distinct types of realism in the philosophy of
mathematics. Realism-in-ontology is the view that the subject matter of
mathematics is the realm of objects that exist independent of the mind,
conventions, and language of the mathematician. Most advocates of this
view hold that mathematical objects -- numbers, functions, points, sets,
etc. -- are abstract, eternal, and do not enter into causal
relationships with material objects. Because of this,
realism-in-ontology is sometimes called platonism.
"Realism-in-truth-value is the view that unambiguous assertions of
mathematics are non-vacuously true or false, independent of the mind,
language, and conventions of the mathematician. (This would seem to be
close to the view that you, Bruno, espouse.)
"There is a natural connection between the two varieties of realism.
Consider the following statement:
'There is a prime number greater than 1,000,000.'
"The realist-in-truth-value holds that this is an objective truth. But
what does it mean? Prima facie, '1,000,000' is a singular terms, and
'prime number' is a common noun. If the surface grammar of this sentence
reflects its logical form, and if 'there is' means 'there exists', then
the sentence entails that both the number 1,000,000 and a greater prime
number exist. For the realist-in-truth-value, this existence is
objective, and so we are led to realism-in-ontology. In sum, if one is a
realist-in-truth-value, then realism-in-ontology is the result of taking
mathematical assertions at face value."
Other references that I have looked up, such as entries in the Stanford
Encyclopedia of Philosophy on "Realism" and "Platonism in the Philosophy
of Mathematics", say similar things. Though, of course, there are
probably more nuances in the understanding of mathematical realism than
there are philosophers of mathematics.
Given the above references, I think it should be clear why I say
"realism or platonism", and refer to "an independently existing
mathematical realm". In Western philosophy at least, that is what
realism in mathematics means -- although things might be different in
Gallic philosophy.
It seems that your idea of arithmetical (mathematical) realism is
entirely from classical logic and is, therefore, essentially a
'realism-in-truth-value' understanding. It is interesting, in that case,
that you make no reference to mathematical objects. You claim that the
truth of propositions such as '2+2=4' is independent of the mind,
language, and conventions of arithmetic, as in the definition of
'realist-in-truth-value' above. But you do not seem to go the additional
step of saying that mathematical objects, numbers and so on, are objects
that actually exist (which would be a form of platonism). If you want to
reject platonism, and the idea that mathematical concepts are objects
that actually exist -- that there is a mathematical realm of objects
that exist independently of any physical existence -- then I suppose you
are entitled to any view that you wish to hold. But you cannot claim
that any such view is uniquely necessary.
If you reject platonism, it is hard to see how you can make sense of
claims such as "All calculations exist in arithmetic", or that physics
arises from the statistics of computations in the universal dovetailer.
Since I reject all forms of arithmetical realism, particularly
platonism, I do not think that your arguments for 'comp' have any merit.
However, the philosophy of mathematics is not an area in which I have
had any particular interest, so apart from rejecting mathematical
realism and platonism, I do not have any strong views about which of the
many alternatives on offer might be an acceptable philosophical attitude
to arithmetic.
Bruce
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
