I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
Georg CantorRead
The essence of mathematics lies precisely in its freedom.
Interpretation
Mathematics embodies freedom in creativity and thought, allowing for exploration and abstract reasoning.
Georg Cantor's quote emphasizes that the true essence of mathematics is found in the freedom it provides to think, create, and explore ideas without restraint. This freedom not only fosters innovation within the mathematical field but also reflects the limitless possibilities of thought processes and problem-solving inherent in mathematics, highlighting its beauty as a discipline that transcends restrictiveness.
In practice
A professor gave a lecture on how mathematical freedom allows for innovative theories in his classroom.
I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
A set is a Many that allows itself to be thought of as a One.
The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.
There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.
The essence of mathematics lies in its freedom.
If we want to travel into the future, we just need to go fast. Really fast. And I think the only way we're ever likely to do that is by going into space.
Nature - how, we don't know - has technology that works in every living cell and that depends on every atom being precisely in the right spot. Enzymes are precise down to the last atom. They're molecules. You put the last atom in, and it's done. Nature does things with molecular perfection.
We only have to capture 1/10,000th of the solar energy landing on earth to completely satisfy all our energy needs.
It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
It is mistaken to claim that global problems will be solved more quickly if only researchers would abandon their quest to understand the universe and knuckle down to work on an agenda of public or political concerns. These are not 'either/or' options - indeed, there is a positive symbiosis between them.
Innovation is everyones responsibility, not just R&Ds.
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