I believe it was Poincare who said, “It is not necessary that a theorem be true, but it is necessary that it be beautiful.” At first sight this seems to be an odd thing to say ; for, surely, the whole practical value of a theorem lies, not in its appearances, but in its truth.
But perhaps his mind was working in a different mode from the practical ; for didn’t he also say that a scientist does not study nature in order to make use of it, but because it is merely beautiful. Also I am sure the idea would have crossed his mind that truth is an opinion ; and our opinions on what is true change over time. Sometime many years, or even centuries can lapse before a theorem (or, more strictly, a hypothesis) may be properly tested for truth. It will be remembered that Aristarchos of Samos argued the hypothesis of helio-centricity in the third century BC.
And then there is the principle of convention. For a theorem to be true, its rationale must be argued by agreed rules of reasoning ; and here, the rules also change over time. Pre-Socratic reasoning is very different from our own – as Socrates himself discovered at the cost of his life. Such reasoning is still current among many peoples, including modern people in the West.
On the other hand, nature does possess beauty, as poets, artists, scientists and people from all sides will testify. Therefore a beautiful theorem, provided it is reasonably grounded, will be very likely true, whether proofs be available or not.
But what makes a thing beautiful? And isn’t beauty also an opinion? Here we are on grounds that are similar to those occupied by reason ; grounds in which convention plays a major part. In very general terms, beauty is evidenced by such qualities as symmetry and proportionality – in such things as form and force, mass and motion, colour and sound.
And our ideas of beauty also change over time. The beauty of an ancient Egyptian portrait or statue does not quite match our own tastes ; and an Aztec painting is something of an acquired taste, too – as is a Salvador Dali portrait.
This raises the interesting question, Can an ugly theorem, that stands to reason alone, be accepted on the ground that it might one day be deemed beautiful?
From all this we can see why truth and beauty have always featured highly in our understanding of nature. And we notice that it is our understanding that we are considering – not that of animals or aliens.
Thank you Jamie
You have succeeded once again to prove that all is not lost and in the process also proved to me that there is some value in science; if it can admire and develop a beautiful theorem about our understanding [your closing line did it all], I will add Poincare’s name to my list for bedtime stories.
As for ever understanding it or that it will always remain true when nothing in the universe remains the same; well, I am not so sure about that one but I shall give it some thought from the starting point of it being beautiful.
That covers it. My knowledge may not have increased greatly in the past year but my tolerance and quest for beauty has. Life, regardless of what we read or see in the Media, has become more beautiful after one more year of it. Maybe I shall add some thoughts on that later.
Hello Ike,
It’s good to see you and be cheered once again by your own cheerfulness. I enjoy reading the thoughts of those scientists who flourished in the hey days of science itself, because almost all of them did thir work out of their love of nature, and not as a mere job, not simply for money. This, I think, kept their minds open so that they did not fall into the trap of materialism. They were content to know that we could never know everything – yes, even the atheist ones (like Poincare, I believe) were glad to study beauty for its own sake.
Good morning Jamie, you raise some interesting points.
“our ideas of beauty also change over time”
This is certainly true in my case. Beautiful (and at the same time proven to be true) were mathematical equations. One such equation is a statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side.
Example of this equation: 2+3=5.
As I have grown older, while still acknowledging the truth in these equations, these statements appear to me as ugly as a Peter Howson painting.
Conversely, some wrongs are stunning in their blatant out of the ordinariness.
Example: 2+2=5.
Absolutely gorgeous.
Enjoy the rest of your Sunday Jamie.
JW
Hello JW,
Ah, the mystery of numbers! The faith we place in them is remarkable when we consider that, no matter how hard you look, nor how far, you will never find a number in nature. Are the stars numbered? are the hairs on our head? We have invented numbers as symbols for we know not what.
But, as you say, for those who have the mind for them, equations are beautiful ; the sort of things you can fall in (or out) of love with – without the expense!