Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 11/19/04
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Date: Fri, 19 Nov 2004 19:53:48 GMT
"Stephen Harris" <cyberguard1048-usenet@yahoo.com> wrote in message
news:4zrnd.46389$QJ3.16198@newssvr21.news.prodigy.com...
>
> "Eray Ozkural exa" <examachine@gmail.com> wrote in message
> news:320e992a.0411190852.7ce02e6c@posting.google.com...
>> I have had such a discussion with an extremely intelligent and
>> experienced mathematician. He told me that PCs are not Turing
>> Machines, because they have "an infinite tape". I think he did not
>> know anything about descriptive complexity. This infinite portion of
>> the tape consists entirely of blank symbols, and therefore has
>> descriptive complexity O(1), which is easily realized by a physical
>> system. When I told him about Ullman's indefinite growing argument, he
>> objected "But when the universe is filled up, it cannot grow any more!
>> Then, it is not infinite", to which I responded "Yes, but there is
>> *nothing* that is larger than the universe." To assume the contrary
>> would be theology, which I despise.
>>
>
> Turing Machines are idealized which means they are not physically
> realized.
> TMs are not meant to have physical constraints applied to them. That mixes
> categories. Sometimes the question is asked, how many sentences can be
> generated in some natural (say English) language? There are finitely many
> words in the language, but the standard answer is that there are countably
> infinite number of sentences, due to appending etc.
>
> These kind of questions ask what is the potential in theory, not what is
> practically possible. Observations like: a person can only articulated
> finitely
> many sentences in a lifetime, or any sentence has to be uttered before
> somebody dies or a machine wears out, or that how many sentences
> potentially exist is related to how many people generate sentences over
> the lifetime of humanity in the universe are not relevant, because that is
> not the question being asked. Every process within the universe is finite
> due to heat death of the universe, so that makes all such questions
> trivial,
> if one interprets them to mean or apply to a physical reality. A Turing
> Machine or potential sentence of a language (there is no pre-existing
> specification that the sentence has to be of finite length) is not of this
> world.
>
> The set of natural numbers is countably infinite and is has some use
> theoretically. Would you claim infinite sets have no use because they
> have more members that there are particles existing in the finite
> universe?
>
> And the original description of a Turing Machine. It is common to call
> this tape 'infinite' though some prefer finitely unbounded. There is no
> physical time constraint applied to when the calculation has to be
> completed. So there are calculations that a physical PC the size of
> a galaxy could not complete before the universe ran out of power to
> energize the computer. A Turing Machine can of course complete
> such a calculation (because the calculation does not need to be infinite,
> just finitely larger/longer in time that can be accomplished by any
> physical device during the existence of the physical universe) because
> the constraint of physical time is not applied to idealized situations.
> Keeping those categories seperate, the idealized and the physical,
> is definitional. The answer to theoretical questions is trivial and
> obvious
> if you mix these categories. Mathematicians invented infinity without
> the requirement that it be physically realized because it was useful.
> Pure mathematics invents formal mathematical systems with no
> requirement that this formal system represent any physical event or
> process. Eray wrote:
>
>> "Yes, but there is
>> *nothing* that is larger than the universe." To assume the contrary
>> would be theology, which I despise.
>
> When you say *nothing* you mean no physical something. Mathematical
> objects need not be physical. Ideas may be generated physically, but the
> idea of a unicorn can exist without the idea being physically manifested.
> The mathematical idea of a circle exists. We do find physical objects
> which remind of this mathematical idea. Pi is the ratio of a circumference
> of a circle to the diameter. Even if you think of Pi as finitely
> unbounded,
> there is still no last digit of Pi, there is still no last digit of Pi. So
> in theory
> you can talk about the digit expansion of Pi after the decimal to a value
> say, 10^10^10^10^ and so on to say millions of exponents of exponents
> and this finite number will exceed the particles in the universe and no
> computer could calculate within the lifetime of the universe.
>
> That does not make Pi theology. You will find Pi used in Physics.
> You will find infinity used in quantum theory which makes theoretical
> predictions which match experiments to 10^11 of real world accuracy.
>
> Mathematics has nothing to do with theology. It certainly does not require
> one to adopt mathematical platonism, a metaphysical realm outside the
> universe.
> They do say that mathematics is 'unreasonably effective'. Mathematics
> starts
> with observations of physical reality and then regularities are then
> *represented*.
> Mathematics is a logical relationship to reality, it is
> idealistic/symbolic,
> especially when formalized, and is a useful tool to predict the behavior
> of
> physical reality.
> It is _not_ the same as physical reality. And that is why concepts of
> mathematics can have theoretical objects; mathematics as abstract thinking
> is not required to map one-to-one to existing physical events or objects.
>
> I can imagine the successor function which adds one to the previous number
> and which can generate the naturals 1,2,3,... and so on and so on into
> infiinity,
> even though I cannot mentally grasp infinity. But you are trying to make
> this
> analagous to grasping God or theology. I cannot visualize God as having a
> 1,2,3... foundation successor function so therefore having an abstract
> existence.
>
> When you made this comparison, infinity and theology/God, to the size
> of the physical universe, you crossed over from debating potential vs.
> actual infinities to declaring abstract thinking is just theology in
> another
> guise.
>
>> "Yes, but there is
>> *nothing* that is larger than the universe." To assume the contrary
>> would be theology, which I despise.
>
> Statements like this are going to appear to others as displaying gaps
> in your background education. Abstract thinking in mathematics does
> not assume that there is a physical object under discussion so that
> _size_ ("larger than the universe") is a pertinent factor. And it is
> muddled to conflate the inability to grasp how a potential infinity
> transforms into an actual infinity as a theological issue. Your statement
> attacks mathematics using even the abstract concept of a *potential*
> infinity as religious mumbo jumbo. A potential infinity is larger than
> any aspect contained within the universe also. And actualized infinity
> certainly has nothing to do with that infinity being manifested within
> the physical universe.
>
> Actualized infinity is another abstract mathematical construct
> conceptualizing completing a potential infinity, neither of which
> abstractions have physical size.
> Your idea reminds me of, There can't be an acutalized infinity(number)
> because
> it would be too long to fit in the universe and nobody would live long
> enough
> to write it down anyway. Two factors having nothing to do with the
> discussion.
> After writing this, I think you may not have realized this.
> --
> "Mathematics - this may surprise or shock
> some - is never deductive in its creation.
> The mathematician at work makes vague
> guesses, visualizes broad generalizations,
> and jumps to unwarranted conclusions.
> He arranges and rearranges his ideas,
> and he becomes convinced of their truth
> long before he can write down a logical
> proof....The deductive stage, writing the
> results down, and writing its rigorous proof
> are relatively trivial once the real insight
> arrives: it is more the draftsman's work not
> the architect's." - Paul Halmos
>
> SH: Achieving or having a mathematical insight is not the
> same thing as having a religious/theological experience.
> The idea that it is the same thing, is itself, a mystical claim.
>
> Language is abstract and symbolic,
> Stephen
>
>
http://www.disf.org/en/Voci/13.asp
"Against both of these bastions of rationalist 18th century philosophy and
its anti-metaphysical program, Cantor poses the distinction between relative
actual infinity, or "transfinite", as a mathematical notion ( CANTOR, III),
and the "absolute" actual infinity, as a metaphysical and theological
notion, typically attributed to the divine nature, and absolutely
unreachable by pure mathematical knowledge. Unfortunately, Cantor thought
his view on infinity as opposed to the Thomist conception, because of his
insufficient knowledge of Aquinas' thought, together with the insufficient
scholarship of some of his interlocutors. Therefore he was led to believe
that he had to systematically oppose the Scholastic philosophical doctrine
with his conception of actual infinity in mathematics. The necessity of
actual infinity here re-appears in a sense that joins the "Parmenidean"
instance with the "Platonic" instance. The necessity for the existence of
actual infinity is so linked by Cantor with the necessity for its
conceivability (Parmenidean instance), properly in relation to the rigorous
definition of the notion of limits within the analytical calculus and
regarding the definition of Dedekind of "real number" as the limit of a
sequence of "rational numbers" not belonging to the sequence itself. These
two notions in fact imply that, in order to let mathematics be founded on
them in a really auto-consistent way (Parmenidean instance), the indefinite
variation of the finite (potential infinity) requested by the notion of
limit has to suppose the a priori "complete determination" of the domain of
variation (Platonic instance). «There is no doubt that we cannot do without
the variable quantities within the sense of potential infinity; and that
from this can be demonstrated the necessity for actual infinity. In order
that there is a variable quantity in a mathematical theory, the "domain" of
its variability must be, strictly speaking, known ahead of time through a
definition. Thus, the said domain must not be itself something variable,
otherwise every base founded for the study of mathematics would vanish.
Consequently, this domain is a definite set of values, and is thereby
actually infinite» (Cantor, 1886, p. 9)."
SH: It is true that the term "actual infinity" has a historical basis which
is theological. But that concluding sentence does not require that history
nor Platonism. Under discussion is a mathematical domain, not a region of
physical or religious space.
"Cantor poses the distinction between relative actual infinity, or
"transfinite", as a mathematical notion (CANTOR, III), and the "absolute"
actual infinity, as a metaphysical and theological notion, typically
attributed to the divine nature, and absolutely unreachable by pure
mathematical knowledge."
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