61 ideas
20678 | The Scientific Revolution was the discovery of our own ignorance [Harari] |
20686 | For millenia people didn't know how to convert one type of energy into another [Harari] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
20692 | Money does produce happiness, but only up to a point [Harari] |
20663 | If a group is bound by gossip, the natural size is 150 people [Harari] |
20677 | Since 1500 human population has increased fourteenfold, and consumption far more [Harari] |
20688 | People 300m tons; domesticated animals 700m tons; larger wild animals 100m tons [Harari] |
20674 | The Nazi aim was to encourage progressive evolution, and avoid degeneration [Harari] |
20679 | We stabilise societies with dogmas, either of dubious science, or of non-scientific values [Harari] |
20690 | The state fostered individualism, to break the power of family and community [Harari] |
20689 | In 1750 losing your family and community meant death [Harari] |
20681 | The sacred command of capitalism is that profits must be used to increase production [Harari] |
20682 | The main rule of capitalism is that all other goods depend on economic growth [Harari] |
20683 | The progress of capitalism depends entirely on the new discoveries and gadgets of science [Harari] |
20687 | In capitalism the rich invest, and the rest of us go shopping [Harari] |
20685 | No market is free of political bias, and markets need protection of their freedoms [Harari] |
20693 | Freedom may work against us, as individuals can choose to leave, and make fewer commitments [Harari] |
20691 | Real peace is the implausibility of war (and not just its absence) [Harari] |
20684 | Financing is increasingly through credit rather than taxes; people prefer investing to taxation [Harari] |
20675 | The more you know about history, the harder it becomes to explain [Harari] |
20676 | History teaches us that the present was not inevitable, and shows us the possibilities [Harari] |
20671 | In order to explain both order and evil, a single evil creator is best, but no one favours that [Harari] |
20664 | Animism is belief that every part of nature is aware and feeling, and can communicate [Harari] |
20666 | Most polytheist recognise one supreme power or law, behind the various gods [Harari] |
20667 | Polytheism is open-minded, and rarely persecutes opponents [Harari] |
20665 | Mythologies are usual contracts with the gods, exchanging devotion for control of nature [Harari] |
20669 | Dualist religions see everything as a battleground of good and evil forces [Harari] |
20670 | Dualist religions say the cosmos is a battleground, so can’t explain its order [Harari] |
20673 | Manichaeans and Gnostics: good made spirit, evil made flesh [Harari] |
20668 | Monotheism appeared in Egypt in 1350 BCE, when the god Aten was declared supreme [Harari] |