FOM: (mercifully brief) reply to Hersh's reply

Reuben Hersh rhersh at
Mon Mar 23 16:42:01 EST 1998

 On Mon, 23 Mar 1998, Randall Holmes wrote:

> (Hersh says) 
> To talk about real mathematics in the proper ideal eternal
> immaterial insubstantial sense, we must evidently rise out of
> or above our own being (as humans) which is physical, material
> and social, cultural.  We, physical, material, social, cultural
> beings must contact and interact with and in some Godelian sense
> actually perceive the transcendental abstract reality of real
> math.
> (end quote)
> We actually interact with structure of the kind that I think
> mathematics is about all the time in everyday life.  There's nothing
> especially mysterious about it.  Moreover, people (even those who
> profess to be materialists) talk about "abstractions" all the time,
> and appear to know what they are talking about :-)
	There's nothing especially mysterious about it as long as
	we don't insist that numbers, unlike us, are abstract,
	and we, unlike numbers are concrete.  Maybe it will
	clarify the point if we go back to Descartes.  He says
	there are two "substances" of different nature, of
	which respectively our bodies and souls partake (say
	mind instead of soul if you like.)  Then someone could
	ask him, "In that case, how do they interact?"  He
	could have said, like you, "No problem we do it all the time."
	But it is a problem, and he recognized it as such, though
	he couldn't solve it (the mind-body problem, we call it nowadays.)
	If you're Marvin Minsky, you can say it's a non-problem.
	Since you're not Marvin Minsky, perhaps you can acknowledge
	the problems, both of mind-body, and of abstract math--concrete

> On any kind of qualification of "being", see below.
> (Hersh says)
> In other words, you can be religious or materialistic,
> but not half and half.
> (end quote)
> The alternative to "materialist" is not "religious".  One can be a realist
> (as opposed to a nominalist) without being religious in the least.
	I am not anti-religious.  Some of my best friends are.  My
	grandfathers were.  I take it that for you "realist" means 

> (Hersh says)
> I think that it is very clear that existence comes in different
> senses or levels.
> (end quote)
> And I think that it is very clear that it does _not_ :-) Things either
> are or they aren't, as it were (in the final analysis).  

	I can't believe you're denying the crucial distinctions
	between the physical, mental and social.   I take it
	rather that you're saying they don't interest you, or
	are unimportant, or can be ignored if one wishes to do so
	in an appropriate logical framework.  Since that is
	exactly the issue that concerns me, perhaps we simply
	don't have any common interest.

Reference of
> ostensibly naming terms does come in different flavors: for example,
> on my own recent account the expression "the number 7" does not refer
> to any specific object; it refers to an object in an (implicitly
> understood) model of arithmetic (or set theory, if one wants to use it
> to count elements of a set).  The unqualified existential assertion I
> would be making is "there are models of arithmetic (or set theory)";
> in any such model a "number 7" can be found.  Similar phenomena of
> context dependent reference are found in ordinary language, of course.
> (Hersh says)
> Then if we have math being real, and we want to understand how that
> can be, we have five choices:
> 1) physical    
> 2) mental (individual subjectivity)   
> 3)  social (intersubjective)   
> 4) transcendental-abstract        
> 5) it's a non-problem, doesn't interest  me, leave it to the philosophers.
> No one has advocated 2.  I have tried to deal with 1.  4 and 5 are
> obviously tenable.  It seems to me both involve a certain amount
> of self-deception and wishful thinking.
> (end quote)
> I'm not a dualist; I recognize alternatives 1 and 2 as variations of
> the same thing (leaving out the word "subjective").  I regard 3 as
> incoherent (we can't really "make things up", even by mutual
> agreement

	This parenthetical remark is crucial.  Why can't we?
	What things do you mean?  Isn't our whole material,
	intellectual, ideological culture something "we"
	"made up"?  Probably you only mean, we can't make
	up anything in math.  This is just a restatement
	of your realist position, presented as if it were
	a known matter of fact.  But it's a controversial matter
	of argument.

; nor can we make them up all on our own, thus my stricture
> on the word "subjective" under alternative 2).  The definition of 4
> doesn't resonate with me; "transcendental" is probably (from Hersh) a
> term of abuse
	not so.  I use it because I think it's more accurate and
	honest than "abstract."  No abuse intended.

 in any event I would not use it.  "abstract" (used of
> universals) presumes a certain philosophical view of their nature
> which I probably don't hold.  But universals (properties, relations)
> can live under heading 4, I suppose.
> Like Hersh, I really don't think 5 is an option.
> Please note that in my recent posting outlining a proposal for the
> nature of the reference of mathematical statements, the only
> requirement I placed on the objects whose actual existence is needed
> to underpin the reference of mathematical assertions is that there be
> infinitely many of them (this requirement may be met in the physical
> universe, which is, so far as anyone can tell, infinite in extent).
> There were no assumptions about "transcendental", "immaterial", etc.
> qualities; in fact, no assumptions about the individual character of
> these objects were needed at all. I specifically noted that they might
> be physical.  Certain second-order assumptions might lead to one
> assuming that the number of these objects was too large to fit in
> the physical universe we are familiar with, but such assumptions were
> not necessary to my account.
> Since I admitted second-order quantification, I suppose I'm reifying
> properties of and relations on these objects.  Universals are not
> generally regarded as material objects, it is true.  It is also true
> that actual reference to universals is a _universal_ feature of daily
> speech of ordinary people.  The materialist (nominalist) position on
> this would be, I suppose, that statements involving references to
> universals can always be rephrased so as to avoid such references; no
> adequate proposal has actually been given for eliminating such
> references (Quine thought about it and concluded that it was
> impossible or at least much harder than nominalists claim), and they
> proliferate as much in the speech of materialists as in the speech of
> the rest of us.  Things one cannot avoid talking about have a strong
> claim to being real.

	"cannot avoid talking about" is too vague for me.
	Quine's dictum referred only to hard science.  When
	questioned about his philosophy of life, he didn't
	allow it any claim to be real.  For him, only the logically
	precise deserved consideration.

> This is longer than I hoped when I wrote the subject line,> 

> And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
> the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
> slow-witted and the deliberately obtuse might | holmes at
> not glimpse the wonders therein. |
	Has it occurred to you that this slogan can be interpreted:
	"Any who are skeptical about the wonders of Cantor's paradise 
	are slow-witted and deliberately obtuse."

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