Futa Helu Lecture - "Mathematics As Art"
Futa Helu
Mathematics As Art
VIDEO SOURCE: Mathematics as Art: A Seminar at Atenisi University by Professor Futa Helu (1998- 1hr 30mins)20
Transcript:
Presenter:
Tonight our opening speaker for this term will be Professor Futa Helu and as
you can see on the board I have his topic for tonights’ discussion “Mathematics
As Art”. Ladies and gentlemen without any further ado let me introduce you to
Professor Helu.
Professor Helu:
The title “Mathematics as Art” is meant more to emphasize the other formulation
of that topic, namely that mathematics is not a science. Whether I can
demonstrate if it is an art or not is a different matter. We will see about
that as we go but I will try and present general arguments and considerations
which would show that mathematics is not a science in a strict sense of the
term ‘science’. Secondly, that the theory I am presenting tonight is exploratory
and tentative. It might stimulus further inquiry into the subject, it may not,
but I hope it will have its value in stimulating thought. Traditionally, well
in the recent past, mathematics has been taken as a science or queen of
sciences. I will be arguing it is a model for science, not a science itself.
The first general
consideration here is that a science is generated by observation, by a mass of
observations and it is born of observation, of investigation into things where
we collect a whole amount of data and we analyse the data and we organize our
thoughts in relation to the observed data, this is science. We formulate
hypothesis and the observation are matched against the hypothesis and this matching
we call ‘verification’. We have the laboratory which can be regarded as a
factory for verification. And if we take one of the standard examples of the
scientist, the biologist, or the botanist, the botanist would be spending his
time among trees and plants, measuring and observing the behaviour of plants
and plant material and he would be analysing all sorts of data relating to
plant and so on, and he would be making generalizations on top of this that is
why science has been come to be regarded as an inductive business because of
the collection of data and pulling together of observations and so on.
Now mathematics has
been divided of course into two main fields of pure mathematics and applied
mathematics. I’m talking about pure mathematics. In pure mathematics there is
very little observation in the sense that we associate that activity with
science, with any of the special sciences. In mathematics the mathematician
sits down in his office or where ever with pen and pad and he works out
equations, formulae, and relationships between amounts, between numbers,
between quantities or between distances and so on. What does he observe? Well,
he does a lot of introspection, of arguing in his head, but using logical principles
and he then arrives at his propositions. So one could say that in mathematics
we generate the whole edifice of mathematical theories by deduction. One can
contrast that with the scientific approach which is through observation of
things but in mathematics we deduce propositions from other propositions.
What about applied
mathematics? Applied mathematics is the use of mathematics to help us solve our
physical problems, scientific problems, and this means we must admit that the two
fields, the sciences and mathematics, which is I said is not a science in the
same sense they interact and they support each other in so many ways. I’m
mentioning this, the case of applied mathematics, because the fact that the
propositions of mathematics apply to the so called ‘real world’ has misled some
mathematicians and philosophers, philosophers of science, into thinking that
mathematics is empirical in the same
sense as the other sciences including my old teacher John Anderson. He many
times says “mathematics is applicable to things” and from that he drew the
conclusion that mathematics is a science, is an empirical science. My view is
that mathematics and the sciences interact. There is a relationship between
them and we cannot stop them interacting in different ways. I will say
something about this a little later. But I mention that because I do not regard
applied mathematics as mathematics in the true sense. Mathematics for me, is
pure mathematics, is a deductive activity. I want to desist from calling it a
deductive science but a deductive art if you like.
Now the second point I want to make here is that for mathematics; form, and symmetry, and rhythm are essential. Form is so pure and is the essence of mathematics. In fact, it is so pure that essentially the logical form of a mathematical formula is “A is A”. Now take for example the formula for area of the circle, which is 'A=π *r^2'. In symbols putting it down like that you might say this is 'A' and this is 'π *r^2', there is no symmetry but if we substitute for 'A' it would be 'π *r^2 = π *r^2', that’s what the equation says. In other words, written fully the equation is “A is A” and all mathematical formulae are of that form. That is why Wittgenstein said that mathematical formulae are tautologies, nothing but tautologies, that the subject predicate are the same. It’s the same with any formula whatsoever.
Now compare this with music.
Nietzsche said in The Birth of Tragedy
that all arts aspire to the condition of music. Well, what is the condition of
music that all arts aspire to? His answer is; form, pure form. Painting,
sculpture, architecture, whatever art you can think of is according to in his
terminology approximating the condition of pure form which is found in music. The
pure forms of course are made up of points in space and time but these points
are embodied by tones that is by sounds.
So music and
mathematics are the same or very similar in this respect that they are both
pure forms. Pure mathematics and music - the pure forms. And again that
reinforces the title that mathematics is an art or more of an art than a
science because of its pure form. The question comes to mind at once, does math
or music have any content? My answer is this ‘yes’. The content of music is
feeling. The content of mathematics is ideas or thoughts or in the language of
medieval philosophers’ concepts/universals, that is things in the mind. ‘Feelings’
are in the mind also but they are different kinds of things. So both the
content of music and mathematics are mental things.
Now I want to remind
you of Platos’ theory of Forms. There’s extent, no account of how Plato came to
his theory, there are historians, classicists, who believe that Plato got it
from Socrates and Socrates got it from the Pythagoreans. That may be the case but
Plato of course was a mathematician. First and foremost he was a mathematician.
This is how I reconstruct his train of thoughts, he was thinking about numbers
and it is common observation that we see numbers embodied in things. For
example a number ‘2’ is embodied by ‘2 fish’ or ‘2 men’ or ‘2 coconuts’ but
these are embodiments and I’m sure Plato ,being a really sharp thinker, noticed
this distinction between the number ‘2’ and the embodiment of the number ‘2’;
of ‘2 fish’ or ‘2 oranges’. But still he distinguished between ‘2 fish’ and
number ‘2’ and he said number ‘2’ itself is a Form but that he means it is
something that we do not perceive with our senses but only with our mind. Mind
you he did not say that the Forms are in our mind. His theory is that number ‘2’
is something that exists but we perceive number ‘2’ with our mind, whereas the
fish we perceive with our sense, sense of sight, sense of feeling and so on, of
touch. And he called these things that exist but cannot be perceived by the
senses, only by the mind, he called them Forms. Alright, there’s also another
point that has to be made, that is Plato did not think of the Forms as being in
the mind, only perceived by the mind not by the senses. Alright, that’s just
very short. So for Plato then we have things which he called ‘Sensibles’ things
that we perceive by our senses and things that we perceive by our mind which he
called Ideas or Concepts. A concept is a Latin term Universals. Alright, is
there any difference between the number ‘2’ embodied by 2 fish and the number ‘2’
embodied by 2 apples? Platos answer is ‘no’ it’s the same number 2. And he gave
an account which was riddled with all sorts of logical difficulties but I think
that the original thesis can stand that we can distinguish between the Forms
things perceived by the mind only and things perceived by the senses and the
mind. Okay so much for that point about Form. What I am doing is building up a
case for the idea that mathematics is a not a science but an art or at the most
a model for science. I want to comment now on John Anderson, my old teacher his
paper, early paper, ‘Empiricism’ which he wrote in 1927. In this paper Anderson
argues that mathematics is empirical, that is, it is a science and it is a
science based on observation. This is opposed to the thesis that I am
presenting now. For Anderson, we come to know number ‘2’ because we have been
observing ‘2 fish’ and ‘2 apples’ for so long ever since man came on this
planet he has been observing ‘2’ things not the number ‘2’ but 2 things. Then
he abstracted the principle that is number 2 from the 2 things. Therefore
mathematics is an empirical science. Now, I do not deny this account of
knowledge that John Anderson presents he is right on how we come to acquire our
sense of numeration in numbers. We have observed the embodiments of mathematical
entities for ages’ and that is how we have set up our numbers. However, it does
not follow that the numbers did not exist before their embodiments and I think
it is important to notice the difference between Plato and John Anderson. Plato
says the numbers or the Forms are fundamental, are primary, the world of sense
that is the embodiments came after, they are imitations. The primary thing is
the Universal, is the Form. The ‘2 fish’ are approximation or they are
embodiments that is my term embodiments of the principle. Now Plato does not
try to refute Andersons account of how we come to recognize the numbers and I
do not refute also. My version is that I am a little different in the sense
that my position is that both mental entities like the Forms and Ideas and the
material entities, embodiments, they are both real in the same way. They are
real. Now Plato thought that the forms or the mental entities are the only real
things. Anderson, I don’t think he would deny things in the mind, but he tended
in his writings to degrade ideas and things in the mind and mental entities. Now
I do not believe like Plato that mental things are prior to physical things,
and I do not subscribe to Andersons view that somehow things are more primary
than mental things. My position is that they are both real, equally real, in
the same sense but they interact these two classes of things. The Forms in Platos’
language, and things or Sensibles, things that we experience and they interact
in different ways. I would regard this as a more thorough going kind of realism
than either Anderson or Plato who both regarded themselves as realists. Realism
I think we have to affirm that mental things or things in the mind are just as real
as things outside the mind. Lo to say, inside and outside may not be accurate
because everything is objective, inside and outside are relative. Okay,
Anderson went on to criticize both Leibnitz and Russell in the same paper;
Leibnitzs’ concept of ‘pure science’ and
Russells’ concept of ‘forms of externality’ but they fall under the same kind
of criticism, same kind of confusion, that is Anderson didn’t realize that the
conclusion, the last fact of his argument, is fallacious, is not logical, is
illogical, it doesn’t follow that because we derive our knowledge of numbers
from things he went on to say ‘therefore our knowledge of numbers and the
mathematics that is built on it is empirical’ that is an illogical step. While
I say however is that things remind of content of reality which we are never
aware of before. Things than can be regarded a reminders. Reminders of other
aspects of reality. They remind us of numbers. They remind us of relationships.
Which we cannot on the strength of purely intellectual reflections find out but
we use things as crutches on which to stand and make our investigations. And I
reject Platos’ theory that Forms are the only real things. Both Forms and
things are real in the same way.
Finally I want to
mention again the idea that mathematics is not a science but a model for
science. Model in what sense? Well in exactness and accuracy. Sciences are only
approximate, they do not have the exactness and the accuracy of mathematics. If
we call mathematics a science then we will have to explain why it is different
from the other sciences, the other sciences can only be approximates, we can
never get rid of their marginal errors or aspects, of those sciences. Now
sciences investigate situations, actual situations, and so far as we know now
any situation is complex that means we can never come to an end of our investigation
of any situation because it is infinitely complex and therefore cannot be
neatly summed up in a number of finite formulae whether mathematical or
otherwise. So although mathematics is the model for accuracy and exactness we
cannot say that it is a tool for the thorough and exhaustive investigation of
any field because as I have said any field is infinitely complex therefore
cannot be exhausted by any form of investigation.
To tame all that, I
would like to propose that mathematics has a very high component of creativity
associated with it. Creativity that is akin to the work of the artist, it comes
from inside him, his sense of Form, sense of symmetry, and rhythm. Not so much observing
nature and observing things but creativity and use of logical principles, the
principles/laws of truth and falsity to build edifices, mathematical theories,
which are artistic creations in their form and structure. Okay, that’s it.
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