[lit-ideas] Mathematical World 3 "objects"
- From: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- To: "lit-ideas@xxxxxxxxxxxxx" <lit-ideas@xxxxxxxxxxxxx>
- Date: Mon, 3 Jun 2013 15:41:39 +0100 (BST)
The problem of understanding World 3 mathematical “objects”, such as
mathematical problems and mathematical arguments and theorems, is taken by
Popper as showing that it is “generally valid” that there is “direct grasp of
World 3 objects by World 2”. In The Self and Its Brain, Popper seeks to
illustrate this “direct” relationship between World 3 and World 2 by discussion
of a theorem of Euclid.
Popper comments (TSAIB p.548, Dialogue XI):
“Although, of course, there are some World 1 brain processes going on all the
time while World 2 is awake, and especially when it is busy in solving problems
or in formulating problems, my thesis is not only that World 2 can grasp World
3 objects, but that it can do so directly; that is to say, although World 1
processes may be going on (in an epiphenomenal manner) at the same time, they
do not constitute a physical or World 1 representation of those World 3 objects
which we try to grasp.
“Let me illustrate this by discussing Euclid’s theorem, that for every
natural number, however large, there exists a greater one which is a prime
number; or, in other words, that there are infinitely many primes. Certainly,
Euclid had impressed upon his memory (and thus presumably upon his brain) some
facts about prime numbers, especially facts about their fundamental properties.
But there can, I think, be little doubt about what must have happened. What
Euclid did, and what went far beyond World 1 memory recordings in the brain,
was that he visualized the (potentially) infinite sequence of natural numbers –
he saw them before his mind, going on and on; and he saw that in the sequence
of natural numbers the prime numbers get less and less frequent as we proceed.
The distances between the prime numbers get, in general, wider and wider
(although this has exceptions; for example it seems that however far we go,
there are still so-called twin
primes which are separated by just one even number; but these twin primes get
rarer too).
“Now, looking at this sequence of numbers intuitively, which is not a
memory affair, he discovered that there was a problem: the problem whether or
not the prime numbers peter out in the end – whether there is a greatest prime
number and then no further ones – or whether the prime numbers go on for ever.
And Euclid solved this problem. Neither the formulation of the problem or the
solution of the problem was based on, or could be read off from, encoded World
3 material. They were based directly on an intuitive grasp of the World 3
situation: of the infinite sequence of natural numbers.”
We might say the problem that Euclid discovered was inherent in the infinite
sequence of natural numbers, as primes are inherent in this sequence, and so
the question of whether the primes themselves constitute an infinite sequence
is inherent in the sequence of natural numbers. But what Euclid developed was
insight, a kind of depth of understanding of this sequence, which led him both
to formulate the problem and then solve it: and Popper stresses that the
“encoded World 3 material” of a sequence of natural numbers is not such that we
can “read off” the problem or its solution. We could write out a sequence of
natural numbers say 1 to 1000 – but this “encoded World 3 material” would not
‘encode’ Euclid’s problem in explicit World 3 terms: Euclid has discovered
distinct World 3 content within the World 3 content of the sequence of natural
numbers – because he has seen this sequence contains primes, which are a
distinct level of
World 3 content that is not reducible to natural numbers, and he has seen that
distinct content raises the question whether it extends infinitely like the
sequence of natural numbers (of which it is a subset).
Popper argues that this kind of insight, or depth of understanding, cannot be
reduced to a World 1 process but should be understood as the World 2 mind
working directly to grasp World 3 content. The thesis that (at least sometimes
- if not “generally”) there is a “direct” relation between World 2 and World 3
is tantamount to denying that in such cases there is any World 1 mediating
between World 2 and its interaction with World 3 – so, even where the World 3
content that World 2 grasps is related to content that is “encoded” in World 1,
what World 2 grasps may transcend the “encoded World 3 content”.
Popper wishes also to oppose the idea that some World 1 brain state must
‘mediate’ between the World 2 mind and its grasp of a World 3 “object”; and he
also opposes the idea that there must be a World 1 brain correlate for any
World 2 grasp of World 3 content such that the World 1 brain correlate is a
causal determinant of the World 2 activity or is inextricably linked by way of
some one-to-one correspondence. Popper accepts that there can be no human World
2 of mind without a World 1 brain (the destruction of a person’s physical brain
would destroy the possibility of their having World 2 activity) and that all
World 2-World 1 interaction is mind-brain interaction, but he conceives the
relationship between World 1 and World 2 as going beyond any kind of one-sided
dependence of World 2 on World 1. We might admit there is a general and loose
one-sided dependence in that the destruction of a person’s physical brain would
destroy the possibility
of their having World 2 activity whereas a person could lack any higher World
2 activity (say when in a coma) though their physical brain is intact. But
Popper wishes to deny that there is a more intricate and specific one-sided
dependence such that for every specific World 2 mental state there exists a
specific World 1 brain state that determines the World 2 mental state or which
correlates in a one-to-one correspondence – indeed, he wishes to deny a range
of positions which conceive the mind-brain dependence in a way that mind is
always merely epiphenomenal or parallelistic to brain events. While admitting
some mind-brain relations are such that World 2 is merely epiphenomenal or
parallelistic to World 1 brain events [optical illusions may be an example,
where even our World 3 based knowledge of an illusion does not alter its
sense-perception because that is determined by World 1 processing], to avoid
epiphenomenalism or parallelism we must admit
cases where the reverse is true – where the brain state is merely
epiphenomenal or running parallel to World 2 mental activity. To give World 2,
especially its higher conscious functions, any evolutionary rationale, we must
view World 2 as having the power to act ‘downwardly’ on physical brain states –
otherwise World 2 lacks any “survival value”.
That Popper’s interactionism is opposed to any thorough-going epiphenomenalism
is shown in the preceding Dialogue X, when Eccles suggests [TSAIB p.537] that
when World 2 is open to World 3 “by direct action…that in between there is
always inserted a step via World1. [Emphasis added.] This of course is obvious
enough if one is deriving one’s conscious experiences from the coded
representation of World 3 on some material object. Then it is clear that it has
to be perceived through the senses going through all World 1 stages of
reception and transmission.” Popper is alert to the dangers in this way of
looking at things – his subsequent discussion of Euclid is an attempt to show
how World 2 may ‘directly’ grasp World 3 without World 1 mediating and without
this appearing like “clairvoyance”. Here he replies: “It is perfectly true that
in many of the interactions between World 2 and World 3, the brain is involved,
and with it, World 1.
But, especially in many creative acts involving World 2 and World 3, I think
that World 1 is not necessarily involved, or that World 1 is involved as an
epiphenomenon of World 2. That is, something is going on in World 1, but it
depends partly on World 2. (This is the idea of interaction.) …It is perfectly
true that…discovery [“of new problems”] is likely to have World 1 processes
going on alongside it; but not, I would stress, in parallel to it, because the
discovery of something new is a unique process, and I do not think that one can
speak of a parallelism between two unique processes which are not analysable
into standard elementary processes.” [That is, a true parallelism requires a
universal law tying the two processes in a one-to-one correspondence.]
This provides some of the background against which the case of Euclid’s theorem
acquires its importance. But there are other important aspects of Euclid’s
theorem apart from its use to oppose epiphenomenalism. In the case of Euclid’s
theorem, we are looking at World 3 content very much from a rational or logical
point of view and how the mind may be understood as working rationally or
logically with such content: this is key because, as Popper has it, what we
need here is openness of World 1 to World 2 and openness of World 2 to World 3
in terms of content being transmuted as a conscious rational and critical
process – mere indeterminacy of a quantum mechanical kind, or probabilistic
explanation of a physical kind, would not furnish us with any way to understand
World 2 activity, and its handling of World 3 content, as ever consciously
rational or critical. Another important idea in the background here is that the
self, and the fully
self-conscious mind, is “anchored” in World 3:- without World 3, for example a
World 3 theory of ourselves as selves and as beings that have been born and
that will die, there cannot be full consciousness of self – of a self that
stretches long before and after immediate awareness. The higher functions of
World 2 are thus dependent on World 3. And this moves us further away again
from trying to understand World 2 as merely an upshot of World 1 brain states.
Popper’s discussion of Euclid’s theorem continues [TSAIB, p.549]:-
“The solution of the problem is that, if we assume that there is a greatest
prime number, then, with the help of this alleged “greatest prime number” we
can construct a greater one. We can take all the prime numbers up to the
“greatest”, multiply them all, including the “greatest”, and then add the
number one. Let us call the number so produced N. We can then show that N must
be a prime number, under the assumption that the factors of N-1 were all the
primes in existence. For if we divide N by any of these factors, the remainder
is one. Thus if N is not prime, it can have only divisors that are greater than
the number which we assumed to be the greatest prime. [Thus N must be a prime
and a larger one than “the greatest prime”.]
“The problem whether there exists a greatest prime number is thus solved,
negatively. The related problem whether there exists a greatest pair of twin
primes has not, to my knowledge, been solved so far.
“Euclid’s proof operates with the following ideas: (1) A potentially
infinite sequence of natural numbers. (2) A finite sequence (of any length) of
prime numbers. (3) A possibly infinite sequence of prime numbers. Euclid
discovered the problem whether the sequence of prime numbers is finite or
infinite; and he solved the problem by discovering that the first of these
alternative leads to the second, and thus to absurdity. No doubt, he operated
with intuitive symbolic representations and diagrams. But these were merely a
help. They neither constituted the problem nor its solution. We may say that
the very idea of infinity – a World 3 idea – cannot have a direct brain
representation, although the word “infinite” may of course have one. This can,
of course, be achieved only by becoming familiar with the World 3 situation and
its various aspects.”
In the above passage, Popper uses the fact that there can be no physical
analogue of infinity in the brain [the brain being finite in physical terms] to
argue that the World 2 grasp of the World 3 idea of infinity transcends World 1
in a way where World 2 and World 3 cannot here be understood as epiphenomenal
to any World 1 brain state.
He continues:-
“My point here is that there need not be a World 1 representation of a
World 3 idea (for example, a model in terms of brain elements) in order that we
can grasp the World 3 idea in question. I regard the thesis of the possibility
of a direct grasp of World 3 objects by World 2 as generally valid (and not
only for infinite World 3 objects like infinite sequences); yet the example of
infinite objects makes it, I think, quite clear that no World 1 representation
of the World 3 object need be involved. We could, of course, build a computer
programmed for an operation (such as adding 1 to any intermediate result) which
goes on for ever. But (1) the computer will not in fact go on for ever but will
wear out (or absorb all the available energy) in a finite time and (2) it will,
if so programmed, deliver a sequence of intermediate results but not a final
result; it is we who interpret the sequence of intermediate results as an
infinite sequence, and
understand what this means. (There are no (finite) physical models or
representations of the World 3 idea of potential infinity.)
“The argument for the direct grasp of World 3 objects does not depend on
the non-existence of World 1 representations of infinity. The decisive point
seems to me this: in the process of discovering a World 3 problem – say a
mathematical problem – we at first vaguely “sense” the problem before it is
formulated either in spoken or in written language. We first suspect its
existence; then we may give some verbal or written indications (epiphenomena as
it were); then we may put it more clearly; and then we may put it sharply.
(Only in this last stage do we represent the problem in language.) It is a
process of making and matching, and making again.
“The completed World 3 proof must be critically checked for validity, and
for this purpose it must be put into a World 1 representation – into language,
preferably into written language. But the invention of the proof was a direct
operation of World 2 upon World 3 – certainly with the help of the brain, but
without any reading off of problems or results from brain-encoded
representations or from incarnations of World 3 objects.
“This suggest that all, or most, creative acts of World 2 which produce new
World 3 objects, whether problems or new proofs or anything of that kind, even
though accompanied by World 1 processes, must be other than readings out of
memory and encoded World 3 objects. Now this is very important, because I think
that this kind of direct contact is also the way in which World 2 uses encoded
or incarnate World 3 objects to see directly their World 3 aspects, as opposed
to their encoding. This is the way in which, in reading a book, we transcend
the encoding on the page and get directly to the meaning.”
We might also emphasize that when we speak we make physical sound – but the
physics of sound does not constitute the meaningful content of what we say: and
no World 1 level of explanation is adequate to explain the meaningful content
of what we say. The physics of sound may be essential to speech but it is
epiphenomenal to its World 3 content: and this is obvious because we may, by
convention [such as changing the natural language we speak], alter the World 1
encodement of the language without altering its World 3 content – so that “The
snow is white” and “Die Schnee ist weiss” differ in World 1 terms but not in
terms of their World 3 content. World 1 provides instruments or vehicles for
embodying and conveying World 3 content but that World 3 content is distinct
from any merely World 1 level of the instrument or vehicle that conveys it –
and, at least in cases like mathematics and in the case of propositions, the
World 3 content may be
invariant throughout differing World 1 forms of its expression.
But my previous post indicates that it may be more problematic to view World 3
artistic content – like music – as invariant irrespective of its World 1 form
of expression – for its World 1 expression may be intrinsic to, or inextricably
linked to, its World 3 content. Hence, while propositionally equivalent, “The
snow is white” and “Die Schnee ist weiss” may not be equivalent for certain
aesthetic purposes – for example, in poetry; and lines that may be more or less
equivalent for some practical purposes [“To be or not to be”; “To live or end
it”] may be far from equivalent in aesthetic or dramatic effectiveness . So
while the World 3 content of music is never merely a World 1 issue of physical
sound [physics cannot explain music’s content qua music] and in this sense its
World 3 content is distinct from World 1, nevertheless the World 1 vehicle for
this World 3 content may be inextricably linked with its World 3 ‘meaning’.
However, I have also explained why this may not be Popper’s view – and his may
be a view that sees the World 3 content of music as independent of its World 1
vehicle in the same way the World 3 content of a proposition is independent of
the World 1 vehicle adopted to express that proposition, and so the World 3
musical “object” remains constant or invariant irrespective of its physical
forms. My own suggestion is that we might consider musical and other artistic
objects in various senses – in one sense we may consider musical “objects” [say
symphonies] as invariant irrespective of their World 1 embodiment [this would
be the “object” in World 3.3 terms] and in other senses we may consider a
particular embodiment of such an “object” [this would be a World 3.1 such
“object”]. Kleiber’s recorded version of the Fifth is a World 3.1 “object” and
not the Fifth in World 3.3 or pure World 3 terms: and this distinction between
kinds of “objects” may give rise to issues as to how our understanding moves
between World 3.1 and World 3.3 “objects”, so that deeper appreciation of the
W3.3 Fifth might increase our appreciation of Kleiber’s W3.1 version and vice
versa.
Donal
London
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