21 ideas
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 17867 | If a concept is not compact, it will not be presentable to finite minds [Almog] |
| Full Idea: If the notion of 'logically following' in your language is not compact, it will not be locally presentable to finite minds. | |
| From: Joseph Almog (Nature Without Essence [2010], 02) |
| 17877 | The number series is primitive, not the result of some set theoretic axioms [Almog] |
| Full Idea: On Skolem's account, to 'get' the natural numbers - that primal structure - do not 'look for it' as the satisfier of some abstract (set-theoretic) axiomatic essence; start with that primitive structure. | |
| From: Joseph Almog (Nature Without Essence [2010], 12) | |
| A reaction: [Skolem 1922 and 1923] Almog says the numbers are just 0,1,2,3,4..., and not some underlying axioms. That makes it sound as if they have nothing in common, and that the successor relation is a coincidence. |
| 17872 | Definitionalists rely on snapshot-concepts, instead of on the real processes [Almog] |
| Full Idea: The definitionalist errs by abstracting away from differences cosmic processes, freezing real, dynamic processes in snapshot-concepts. | |
| From: Joseph Almog (Nature Without Essence [2010], 08) | |
| A reaction: You could hardly do science at all if you didn't 'abstract away from the differences in cosmic processes'. We can't write about sea-waves, because they all differ slightly? 'Electron' is a snapshot concept. |
| 17871 | Fregean meanings are analogous to conceptual essence, defining a kind [Almog] |
| Full Idea: Ever since Frege, semantic definitionalists have posited a meaning ('sinn') for a name; the meaning/sinn is their semantic analog to the conceptual essence, as ontologically defining of the kind. | |
| From: Joseph Almog (Nature Without Essence [2010], 07) |
| 17866 | Essential definition aims at existence conditions and structural truths [Almog] |
| Full Idea: The essentialist encapsulating formula is meant to be existence-exhaustive (an attribute the satisfaction of which is logically necessary and sufficient to be the thing) and truth-exhaustive (promising all the structural truths). | |
| From: Joseph Almog (Nature Without Essence [2010], 01) | |
| A reaction: [compressed] If he thinks essentialism means that one short phrase can achieve all this, then it is not surprising that Almog renounces his former essentialism in this essay. He may, however, have misunderstood. He should reread Aristotle. |
| 17868 | Surface accounts aren't exhaustive as they always allow unintended twin cases [Almog] |
| Full Idea: A surface-functional characterisation is not exhaustive. It allows unintended twins, alien intruders with different structures - water lookalikes that are not H2O and lookalike infinite structures that are not the natural numbers. | |
| From: Joseph Almog (Nature Without Essence [2010], 03) | |
| A reaction: He rests this on the claim in mathematical logic that fully expressive systems are always non-categorical (having unintended twins). Set theory is not fully categorical, but Peano Arithmetic is. Almog's main anti-essentialist argument. |
| 17870 | Alien 'tigers' can't be tigers if they are not related to our tigers [Almog] |
| Full Idea: Animals roaming jungles on some planet at the other end of the galaxy with the tiger-look and the tiger genetic make-up but with a disjoint evolutionary history are not the same species as the earthly tigers. | |
| From: Joseph Almog (Nature Without Essence [2010], 10) | |
| A reaction: I disagree. If two independent cultures build boats, they are both boats. If we manufacture a tiger which can breed with other tigers, we've made a tiger. His 'tigers' would scream for explanation, precisely because they are tigers. If not, no puzzle. |
| 17869 | Kripke and Putnam offer an intermediary between real and nominal essences [Almog] |
| Full Idea: Kripke and Putnam offer us enhanced essences, still formulable in one short sentence and locally graspable. They offer between Locke's mind-boggling definitive real essence and his mind-friendly but not definitive nominal essence. | |
| From: Joseph Almog (Nature Without Essence [2010], 04) | |
| A reaction: The solution is to add a 'deep structure' which serves both ends. |
| 17876 | Individual essences are just cobbled together classificatory predicates [Almog] |
| Full Idea: The key for the essentialist is classificatory predication. It is only a subsequent extension of this prime idea that leads us to cobble together enough such essential predications to make an individuative essential property. | |
| From: Joseph Almog (Nature Without Essence [2010], 11) | |
| A reaction: So the essence is just a cross-reference of all the ways we can think of to classify it? I don't think so. Which are the essential classifications? |
| 17873 | Water must be related to water, just as tigers must be related to tigers [Almog] |
| Full Idea: It is a blindspot to say that to be a tiger one must come from tigers, but to be water one needn't come from water. ...The error lies in not appreciating that to be water one still must come from somewhere in the cosmos, indeed, from hydrogen and oxygen. | |
| From: Joseph Almog (Nature Without Essence [2010], 09) | |
| A reaction: A unified picture is indeed desirable, but a better solution is to say that the essence of a tiger is in its structure, not in its origins. There are many ways to produce an artefact. There could be many ways to produce a tiger. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 17864 | Defining an essence comes no where near giving a thing's nature [Almog] |
| Full Idea: The natures of things are neither exhausted nor even partially given by 'defining essences'. | |
| From: Joseph Almog (Nature Without Essence [2010], Intro) | |
| A reaction: A better criticism of essentialism. 'Natures' is a much vaguer word than 'essences', however, because the latter refers to what is stable and important, whereas natures could include any aspect. Being ticklish is in my nature, but not in my essence. |
| 17863 | Essences promise to reveal reality, but actually drive us away from it [Almog] |
| Full Idea: The essentialist line (one I trace to Aristotle, Descartes and Kripke) is driving us away from, not closer to, the real nature of things. It promised a sort of Hubble telescope - essences - able to reveal the deep structure of reality. | |
| From: Joseph Almog (Nature Without Essence [2010], Intro) | |
| A reaction: I suspect this is tilting at a straw man. No one thinks we should hunt for essences instead of doing normal science. 'Essence' just labels what you've got when you succeed. |