220 ideas
21875 | The wisdom of a free man is a meditation on life, not on death [Spinoza] |
17230 | If we are not wholly wise, we should live by good rules and maxims [Spinoza] |
17200 | We must be careful to keep words distinct from ideas and images [Spinoza] |
4840 | Reason perceives things under a certain form of eternity [Spinoza] |
17194 | Reason only explains what is universal, so it is timeless, under a certain form of eternity [Spinoza] |
17213 | In so far as men live according to reason, they will agree with one another [Spinoza] |
4819 | There is necessarily for each existent thing a cause why it should exist [Spinoza] |
21864 | Truth is its own standard [Spinoza] |
8018 | Spinoza's life shows that love of truth which he proclaims as the highest value [MacIntyre on Spinoza] |
5641 | For Spinoza, 'adequacy' is the intrinsic mark of truth [Spinoza, by Scruton] |
4816 | A true idea must correspond with its ideate or object [Spinoza] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
20309 | If our ideas are adequate, what follows from them is also adequate [Spinoza] |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
17185 | Mathematics deals with the essences and properties of forms [Spinoza] |
15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
17222 | The sum of its angles follows from a triangle's nature [Spinoza] |
17197 | The idea of a triangle involves truths about it, so those are part of its essence [Spinoza] |
15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
18176 | Pure mathematics is pure set theory [Cantor] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
17174 | Outside the mind, there are just things and their properties [Spinoza] |
17176 | The more reality a thing has, the more attributes it has [Spinoza] |
17179 | There must always be a reason or cause why some triangle does or does not exist [Spinoza] |
17186 | Men say they prefer order, not realising that we imagine the order [Spinoza] |
20127 | Laws of nature are universal, so everything must be understood through those laws [Spinoza] |
17170 | An 'attribute' is what the intellect takes as constituting an essence [Spinoza] |
17171 | A 'mode' is an aspect of a substance, and conceived through that substance [Spinoza] |
17195 | Things persevere through a force which derives from God [Spinoza] |
17206 | The essence of a thing is its effort to persevere [Spinoza] |
17192 | The 'universal' term 'man' is just imagining whatever is the same in a multitude of men [Spinoza] |
17188 | A thing is unified if its parts produce a single effect [Spinoza] |
5639 | Spinoza implies that thought is impossible without the notion of substance [Spinoza, by Scruton] |
21857 | Substance is the power of self-actualisation [Spinoza, by Lord] |
4813 | Substance is that of which an independent conception can be formed [Spinoza] |
4828 | The essence of a thing is what is required for it to exist or be conceived [Spinoza] |
17187 | Essence gives existence and conception to things, and is inseparable from them [Spinoza] |
17191 | Nothing is essential if it is in every part, and is common to everything [Spinoza] |
17184 | All natures of things produce some effect [Spinoza] |
17205 | Only an external cause can destroy something [Spinoza] |
17175 | There cannot be two substances with the same attributes [Spinoza] |
17173 | Two substances can't be the same if they have different attributes [Spinoza] |
17183 | Things are impossible if they imply contradiction, or their production lacks an external cause [Spinoza] |
4299 | Contingency is an illusion, resulting from our inadequate understanding [Spinoza, by Cottingham] |
4839 | Reason naturally regards things as necessary, and only imagination considers them contingent [Spinoza] |
4824 | We only call things 'contingent' in relation to the imperfection of our knowledge [Spinoza] |
4822 | Divine nature makes all existence and operations necessary, and nothing is contingent [Spinoza] |
17182 | Necessity is in reference to essence or to cause [Spinoza] |
4818 | People who are ignorant of true causes imagine anything can change into anything else [Spinoza] |
20310 | Error does not result from imagining, but from lacking the evidence of impossibility [Spinoza] |
17208 | A horse would be destroyed if it were changed into a man or an insect [Spinoza] |
17209 | A thing is contingent if nothing in its essence determines whether or not it exists [Spinoza] |
5640 | Spinoza's three levels of knowledge are perception/imagination, then principles, then intuitions [Spinoza, by Scruton] |
17211 | Understanding is the sole aim of reason, and the only profit for the mind [Spinoza] |
21801 | Unlike Descartes' atomism, Spinoza held a holistic view of belief [Spinoza, by Schmid] |
21863 | You only know you are certain of something when you actually are certain of it [Spinoza] |
17199 | A man who assents without doubt to a falsehood is not certain, but lacks a cause to make him waver [Spinoza] |
17193 | True ideas intrinsically involve the highest degree of certainty [Spinoza] |
5638 | 'I think' is useless, because it is contingent, and limited to the first person [Spinoza, by Scruton] |
4831 | If the body is affected by an external object, the mind can't help believing that the object exists [Spinoza] |
4865 | The eyes of the mind are proofs [Spinoza] |
20306 | Once we have experienced two feelings together, one will always give rise to the other [Spinoza] |
4835 | Anyone who knows, must know that they know, and even know that they know that they know.. [Spinoza] |
20308 | Encounters with things confuse the mind, and internal comparisons bring clarity [Spinoza] |
4312 | To understand a phenomenon, we must understand why it is necessary, not merely contingent [Spinoza, by Cottingham] |
4833 | The human mind is the very idea or knowledge of the human body [Spinoza] |
16198 | Knowledge is the essence of the mind [Spinoza] |
17196 | The will is not a desire, but the faculty of affirming what is true or false [Spinoza] |
17198 | Will and intellect are the same thing [Spinoza] |
17201 | The will is finite, but the intellect is infinite [Spinoza] |
21805 | Spinoza held that the mind is just a bundle of ideas [Spinoza, by Schmid] |
17204 | Animals are often observed to be wiser than people [Spinoza] |
17212 | To understand is the absolute virtue of the mind [Spinoza] |
21804 | Faculties are either fictions, or the abstract universals of ideas [Spinoza] |
4832 | If the body is affected by two things together, the imagining of one will conjure up the other [Spinoza] |
21869 | Our own force of persevering is nothing in comparison with external forces [Spinoza] |
20307 | As far as possible, everything tries to persevere [Spinoza] |
21803 | The conatus (striving) of mind and body together is appetite, which is the essence of man [Spinoza] |
4836 | The mind only knows itself by means of ideas of the modification of the body [Spinoza] |
21861 | Self-knowledge needs perception of the affections of the body [Spinoza] |
17216 | The poet who forgot his own tragedies was no longer the same man [Spinoza] |
4814 | A thing is free if it acts by necessity of its own nature, and the act is determined by itself alone [Spinoza] |
21802 | An act of will can only occur if it has been caused, which implies a regress of causes [Spinoza] |
4837 | 'Free will' is a misunderstanding arising from awareness of our actions, but ignorance of their causes [Spinoza] |
4843 | Would we die if we lacked free will, and were poised between equal foods? Yes! [Spinoza] |
4844 | The mind is not free to remember or forget anything [Spinoza] |
4311 | We think we are free because we don't know the causes of our desires and choices [Spinoza] |
7828 | The actual world is the only one God could have created [Spinoza] |
21860 | Ideas and things have identical connections and order [Spinoza] |
4308 | Mind and body are one thing, seen sometimes as thought and sometimes as extension [Spinoza] |
4846 | We are incapable of formulating an idea which excludes the existence of our body [Spinoza] |
4834 | Mind and body are the same thing, sometimes seen as thought, and sometimes as extension [Spinoza] |
23951 | Emotion is a modification of bodily energy, controlling our actions [Spinoza] |
23990 | The three primary emotions are pleasure, pain, and desire [Spinoza, by Goldie] |
4849 | The three primary emotions are pleasure, pain and desire [Spinoza] |
4864 | An emotion is only bad if it hinders us from thinking [Spinoza] |
17203 | Minds are subject to passions if they have inadequate ideas [Spinoza] |
7832 | Stoics want to suppress emotions, but Spinoza overcomes them with higher emotions [Spinoza, by Stewart,M] |
4863 | An emotion comes more under our control in proportion to how well it is known to us [Spinoza] |
4841 | People make calculation mistakes by misjudging the figures, not calculating them wrongly [Spinoza] |
20311 | An idea involves affirmation or negation [Spinoza] |
4830 | An 'idea' is a mental conception which is actively formed by the mind in thinking [Spinoza] |
4842 | Ideas are not images formed in the brain, but are the conceptions of thought [Spinoza] |
21807 | Ideas are powerful entities, which can produce further ideas [Spinoza, by Schmid] |
8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
4838 | Claiming that actions depend on the will is meaningless; no one knows what the will is [Spinoza] |
4309 | Spinoza argues that in reality the will and the intellect are 'one and the same' [Spinoza, by Cottingham] |
20305 | Whenever we act, then desire is our very essence [Spinoza] |
21868 | We love or hate people more strongly because we think they are free [Spinoza] |
17202 | We are the source of an action if only our nature can explain the action [Spinoza] |
21865 | We act when it follows from our nature, and is understood in that way [Spinoza] |
21873 | Men only agree in nature if they are guided by reason [Spinoza] |
21872 | We seek our own advantage, and virtue is doing this rationally [Spinoza] |
17189 | The essence of man is modifications of the nature of God [Spinoza] |
17207 | By 'good' I mean what brings us ever closer to our model of human nature [Spinoza] |
8019 | Along with his pantheism, Spinoza equates ethics with the study of human nature [Spinoza, by MacIntyre] |
17229 | If infancy in humans was very rare, we would consider it a pitiful natural defect [Spinoza] |
4845 | We don't want things because they are good; we judge things to be good because we want them [Spinoza] |
4848 | Love is nothing else but pleasure accompanied by the idea of an external cause [Spinoza] |
17217 | Love is joy with an external cause [Spinoza] |
7833 | Spinoza names self-interest as the sole source of value [Spinoza, by Stewart,M] |
17224 | If our ideas were wholly adequate, we would have no concept of evil [Spinoza] |
21870 | Music is good for a melancholic, bad for a mourner, and indifferent to the deaf [Spinoza] |
4860 | Man's highest happiness consists of perfecting his understanding, or reason [Spinoza] |
4847 | Pleasure is a passive state in which the mind increases in perfection [Spinoza] |
4859 | Pleasure is only bad in so far as it hinders a man's capability for action [Spinoza] |
4851 | Reason demands nothing contrary to nature, and so it demands self-love [Spinoza] |
17220 | Self-satisfaction is the highest thing for which we can hope [Spinoza] |
4852 | Both virtue and happiness are based on the preservation of one's own being [Spinoza] |
17210 | All virtue is founded on self-preservation [Spinoza] |
21871 | The more we strive for our own advantage, the more virtuous we are [Spinoza] |
17214 | To act virtuously is to act rationally [Spinoza] |
4856 | To live according to reason is to live according to the laws of human nature [Spinoza] |
17221 | A man ignorant of himself is ignorant of all of the virtues [Spinoza] |
17225 | In a free man, choosing flight can show as much strength of mind as fighting [Spinoza] |
17218 | People who live according to reason should avoid pity [Spinoza] |
17219 | A person unmoved by either reason or pity to help others is rightly called 'inhuman' [Spinoza] |
4857 | Pity is a bad and useless thing, as it is a pain, and rational people perform good deeds without it [Spinoza] |
17223 | Pity is not a virtue, but at least it shows a desire to live uprightly [Spinoza] |
17228 | Rational people judge money by needs, and live contented with very little [Spinoza] |
4853 | Rational people are self-interested, but also desire the same goods for other people [Spinoza] |
4858 | A rational person will want others to have the goods he seeks for himself [Spinoza] |
4855 | If people are obedient to reason, they will live in harmony [Spinoza] |
21874 | The ideal for human preservation is unanimity among people [Spinoza] |
8020 | Only self-knowledge can liberate us [Spinoza, by MacIntyre] |
7412 | Spinoza extended Hobbes's natural rights to cover all possible desires and actions [Spinoza, by Tuck] |
17227 | Slavery is a disgraceful crime [Spinoza] |
17226 | The best use of talent is to teach other people to live rationally [Spinoza] |
4854 | It is impossible that the necessity of a person's nature should produce a desire for non-existence [Spinoza] |
17215 | Animals feel, but that doesn't mean we can't use them for our pleasure and profit [Spinoza] |
17190 | We can easily think of nature as one individual [Spinoza] |
4826 | Nature has no particular goal in view, and final causes are mere human figments [Spinoza] |
1587 | Spinoza strongly attacked teleology, which is the lifeblood of classical logos [Roochnik on Spinoza] |
1588 | For Spinoza eyes don't act for purposes, but follow mechanical necessity [Roochnik on Spinoza] |
12731 | Final causes are figments of human imagination [Spinoza] |
4821 | An infinite line can be marked in feet or inches, so one infinity is twelve times the other [Spinoza] |
17177 | In nature there is just one infinite substance [Spinoza] |
4850 | A final cause is simply a human desire [Spinoza] |
4815 | From a definite cause an effect necessarily follows [Spinoza] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
7835 | The key question for Spinoza is: is his God really a God? [Stewart,M on Spinoza] |
7571 | Spinoza's God is not a person [Spinoza, by Jolley] |
12928 | Spinoza's God is just power and necessity, without perfection or wisdom [Leibniz on Spinoza] |
4823 | God does not act according to the freedom of the will [Spinoza] |
19435 | God is not loveable for producing without choice and by necessity; God is loveable for his goodness [Leibniz on Spinoza] |
7609 | God is the sum and principle of all eternal laws [Spinoza, by Armstrong,K] |
17172 | God is a substance with infinite attributes [Spinoza] |
17231 | God feels no emotions, of joy or sorrow [Spinoza] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
4314 | God is wholly without passions, and strictly speaking does not love anyone [Spinoza, by Cottingham] |
21859 | God has no purpose, because God lacks nothing [Spinoza] |
4825 | To say that God promotes what is good is false, as it sets up a goal beyond God [Spinoza] |
21856 | Spinoza says a substance of infinite attributes cannot fail to exist [Spinoza, by Lord] |
17178 | Denial of God is denial that his essence involves existence, which is absurd [Spinoza] |
21858 | God is being as such, and you cannot conceive of the non-existence of being [Spinoza, by Lord] |
4820 | God must necessarily exist, because no reason can be given for his non-existence [Spinoza] |
17169 | Some things makes me conceive of it as a thing whose essence requires its existence [Spinoza] |
4817 | If a thing can be conceived as non-existing, its essence does not involve existence [Spinoza] |
4827 | Priests reject as heretics anyone who tries to understand miracles in a natural way [Spinoza] |
17180 | Everything is in God, and nothing exists or is thinkable without God [Spinoza] |
17181 | God is the efficient cause of essences, as well as of existences [Spinoza] |
12757 | That God is the substance of all things is an ill-reputed doctrine [Leibniz on Spinoza] |
4829 | The human mind is part of the infinite intellect of God [Spinoza] |
7836 | In Spinoza, one could substitute 'nature' or 'substance' for the word 'God' throughout [Spinoza, by Stewart,M] |
7482 | Resurrection developed in Judaism as a response to martyrdoms, in about 160 BCE [Anon (Dan), by Watson] |
21876 | After death, something eternal remains of the mind [Spinoza] |
7831 | Spinoza's theory of mind implies that there is no immortality [Spinoza, by Stewart,M] |