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The notion that a theorem's usefulness is not solely practical but also lies in its capacity to justify other theorems within mathematics.
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CHAPTER 4 become acquainted with the use of the principle which requires justifying. In other words, before we can reasonably justify a deep lying principle, we first need to be familiar with how it works. We might suppose this practice of deferred justification to be operative elsewhere. It is a notable fact that in mathematics very few useful theorems remain unproved. By ‘useful’ I do not necessarily mean with practical application outside mathe- matics. A theorem can be useful mathematically, for example to justify another theorem. One of the most ‘useless’ theorems in mathematics is Gold- bach’s conjecture. We do not frequently find ourselves saying ‘if only we knew that every even number greater than 2 could be represented as a sum of two prime numbers, we should be able to show that. . .. D J Spencer Brown, in a private com- munication, suggested that their apparent uselessness is not exactly a reason why such theorems cannot be proved, but is a reason for supposing that if a valid proof were given today, nobody would recognize it as such, since nobody is yet familiar with the ground on which such a proof would rest. I shall have more to say about this in the notes to Chapters 8 and 11. Chapter 4 In all mathematics it becomes apparent, at some stage, that we have for some time been following a rule without being consciously aware of the fact. This might be described as the use of a covert convention. A recognizable aspect of the advancement of mathematics consists in the advancement of the consciousness of what we are doing, whereby the covert becomes overt. Mathematics is in this respect psychedelic. The nearer we are to the beginning of what we set out to achieve, the more likely we are to find, there, procedures which have been adopted without comment. Their use can be con- sidered as the presence of an arrangement in the absence of an agreement. For example, in the statement and proof of theorem 1 it is arranged (although not agreed) that we shall write on a plane surface. If we write on the surface of a torus the theorem is not true. (Or to make it true, we must be more explicit.) 85Chapter 4 discusses the principle of deferred justification in mathematics, emphasizing that before justifying a deep principle, one must first become familiar with its operation. It highlights how many mathematical theorems rely on covert conventions that become explicit only through increased consciousness of the underlying assumptions, such as the implicit use of a plane surface in proofs. The chapter also touches on the idea that some theorems, like Goldbach's conjecture, remain unproved not due to impossibility but because the mathematical community is not yet familiar with the foundational ground required for their proof.