A quotes list created by Lee Sonogan
A “scientific charlatan”, a “renegade” and a “corrupter of youth” by his peers, he is a highly underrated man who spoke about infinity and many sets of numbers. You could summarise his work (1845-1918) by making the clear distinction between what are real numbers that are more numerous than natural numbers. From the transfinite number to much more, ‘No one shall expel us from the paradise that Cantor has created’ by a collage against his critics..
- Some infinites are bigger than other infinites.
- “The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series…but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.”
- “One other thing to keep in mind, though, is that Cantor’s transfinite math will end up totally undercutting Aristotelian objections like the above (b) to Dedekind’s proof, since Cantor’s theory will constitute direct evidence that actually-infinite sets can be understood and manipulated, truly handled by the human intellect, just as velocity and acceleration are handled by calculus. So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are.”
- “The transfinite numbers themselves are in a certain sense new irrationals, and in fact I think the best way to define the finite irrational numbers is entirely similar; I might even say in principle it is the same as my method for introducing transfinite numbers. One can absolutely assert: the transfinite numbers stand or fall with the finite irrational numbers; they are alike in their most intrinsic nature; for the former like these latter are definite, delineated forms or modifications of the actual infinite.”
- “In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just thoroughly existent as the finite positive integers.”
- My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
- The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
- I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not – I do not say divisible – but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
- “The essence of mathematics is in its freedom.”
- “In Mathematics the art of proposing a question must be held of higher value than solving it.”
- “My beautiful proof lies all in ruins.”
This is very relevant to what I am currently working on behind the scenes. To read and greater understand these symbols in modern formulas that all share the same meaning. This is also something I coming back to reference again for a similar purpose. No matter how near it always seems, the implies that infinity is bigger than you may think of just a paradox not meant to be known.