17 ideas
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 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) |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 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. |
| 12712 | Substance is that which can act [Leibniz] |
| Full Idea: I define substance as that which can act. | |
| From: Gottfried Leibniz (Definitiones cogitationesque metaphysicae [1678], A6.4.1398), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
| A reaction: This is in tune with the notion that to exist is to have causal powers. I find the view congenial, and the middle period of Leibniz's thought, before monads became too spiritual, chimes in with my view. |
| 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. |
| 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. |
| 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? |
| 12737 | Nature can be fully explained by final causes alone, or by efficient causes alone [Leibniz] |
| Full Idea: All the phenomena of nature can be explained solely by final causes, exactly as if there were no efficient causes; and all the phenomena of nature can be explained solely by efficient causes, as if there were no final causes. | |
| From: Gottfried Leibniz (Definitiones cogitationesque metaphysicae [1678], A6.4.1403), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6 | |
| A reaction: Somewhat speculative (a virtue!), but it is interesting to see him suggesting that there might be two complete and satisfactory explanations, which never touched one another. I can't see Aristotle agreeing with that. |
| 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. |
| 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. |