18 ideas
| 17879 | Axiomatising set theory makes it all relative [Skolem] |
| Full Idea: Axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatisation. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.296) |
| 17878 | If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem] |
| Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293) |
| 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. |
| 17880 | Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem] |
| Full Idea: The initial foundations should be immediately clear, natural and not open to question. This is satisfied by the notion of integer and by inductive inference, by it is not satisfied by the axioms of Zermelo, or anything else of that kind. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.299) | |
| A reaction: This is a plea (endorsed by Almog) that the integers themselves should be taken as primitive and foundational. I would say that the idea of successor is more primitive than the integers. |
| 17881 | Mathematician want performable operations, not propositions about objects [Skolem] |
| Full Idea: Most mathematicians want mathematics to deal, ultimately, with performable computing operations, and not to consist of formal propositions about objects called this or that. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.300) |
| 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? |
| 8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
| Full Idea: Sceptics respond to the regress problem by denying knowledge; Foundationalists accept justifications without reasons; Positists say reasons terminate is mere posits; Coherentists say mutual support is justification; Infinitists accept the regress. | |
| From: James Van Cleve (Why coherence is not enough [2005], I) | |
| A reaction: A nice map of the territory. The doubts of Scepticism are not strong enough for anyone to embrace the view; Foundationalist destroy knowledge (?), as do Positists; Infinitism is a version of Coherentism - which is the winner. |
| 8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
| Full Idea: Contemporary foundationalists are seldom of the strong Cartesian variety: they do not insist that basic beliefs be absolutely certain. They also tend to allow that coherence can enhance justification. | |
| From: James Van Cleve (Why coherence is not enough [2005], III) | |
| A reaction: It strikes me that they have got onto a slippery slope. How certain are the basic beliefs? How do you evaluate their certainty? Could incoherence in their implications undermine them? Skyscrapers need perfect foundations. |
| 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. |