Here are my notes from A MATHEMATICIAN’S APOLOGY by G.H. Hardy. I studied this essay to better understand the nature of patterns, which are described by the Laws of Metaphysics on a grander scale.
Hardy seems to be obsessed with good and evil, which makes him less objective. But here is the essence of his essay.
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It is preferable to do mathematics than talk about it. Satisfaction in mathematics comes from discovering new patterns rather than putting those patterns to use. Mathematics requires the freshness of a young mind. Doing mathematics is very pleasurable and its achievements have certain permanence.
Mathematics is done for its own sake. It is quite something to express mathematical ideas in understandable language. Mathematics consists of patterns of ideas that have permanence. The best mathematics lies not in its practical consequences but in the significance of the ideas which it connects.
Simple examples are (a) Euclid’s proof of the existence of infinity of prime numbers, and (b) Pythagoras’s proof of the ‘irrationality’ of √2.
Mathematical patterns may be judged by their beauty and seriousness. The theorems of Euclid and Pythagoras have influenced thought profoundly, even outside mathematics. Practical applications are concerned only with approximations.
A ‘serious’ theorem contains ‘significant’ ideas that consist of certain generality and depth. Abstraction refers to any pattern of ideas, but generality is marked by how many different ideas are connected. Mathematical ideas may be viewed as arranged in strata. Generality and depth go hand-in-hand.
Mathematical reality lies outside of us. Our function is to discover or observe the continuity, harmony and consistency of its beautiful patterns. The aesthetics of mathematics lies in the creativity and simplicity of the proofs of its theorems.
The physical patterns have their own structure which derives its explanation from the patterns discovered in mathematics. The physical reality is an approximation derived from mathematical reality.
The mathematics of great mathematicians has permanent aesthetic value. That knowledge goes much beyond providing merely the material comfort of mankind.
Real mathematics is the most austere and the most remote of all the arts and sciences. It is not a contemplative but a creative subject. It may be justified as art.
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