Antinomies

Frank and Ernest cartoon, 5KAntinomies are discomforting things. If you haven’t run into them before, they were a topic of debate and discussion introduced into modern philosophy by Kant (Unless he had some forebear of which I’m unaware), though you might say that they bear some resemblance to Zeno’s paradoxes, or that Zeno’s paradoxes are perhaps the product of the antinomies.

Here’s Wikipedia’s explanation of Kant’s First Antinomy:

in the First Antinomy, Kant proves the thesis that time must have a beginning by showing that if time had no beginning, then an infinity would have elapsed up until the present moment. This is a manifest contradiction because infinity cannot, by definition, be completed by “successive synthesis” — yet just such a finalizing synthesis would be required by the view that time is infinite; so the thesis is proven. Then he proves the antithesis, that time has no beginning, by showing that if time had a beginning, then there must have been “empty time” out of which time arose. This is incoherent (for Kant) for the following reason. Since, necessarily, no time elapses in this pretemporal void, then there could be no alteration, and therefore nothing (including time) would ever come to be: so the antithesis is proven. Reason makes equal claim to each proof, since they are both correct, so the question of the limits of time must be regarded as meaningless.

I suspect my own approach is a dumbing down of Kant’s loftier and more metaphysical intentions, but I think of the antinomies as illustrating the limits of reason. We always think that if we can only nut something out properly we’ll get the answer.  Well the antinomies suggest that there are limitations to that – that you may begin with perfectly good concepts, or concepts that are helpful in one domain and end up tangled in paradoxes. Hegel regarded them as an example of the way in which too much wit outwits itself.

And disciplines other than philosophy have pursued their own logic, helpful as it may be in in making practical disciplinary progress all the way to a point where they end in paradoxes. Here are two examples.

The search for logical foundations of mathematics ended in tears, or at least in paradoxes, like Russell’s paradox. The set of all sets that are not members of themselves both contains itself and doesn’t contain itself. Later Gödel’s second incompleteness theorem demonstrated that “For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.” (Wikipedia)

Economics too has its antinomies, though in a far more prosaic way than those above. Perfect competition is probably the best example. Perfect competition is a mathematical construct which takes to its logical conclusion the tendencies that competition produces in the market – leading to price equalling marginal cost.  But the thing that we end up with is a place where there is no incentive to compete!. For the market price is given – so one produces as much as one wants to and then no more. One gains nothing by taking someone else’s market.  Firms may as well co-operate as compete, but there’s no incentive for them to co-operate either as each producer is at the technology frontier, so they have nothing to offer each other. And being at the technology frontier they have no incentive to innovate, and if there was a need to innovate, their production would generate no surplus with which to invest to innovate. For this reason, perfect competition also fails to do one of the most central things that competition does in the real world, which is to motivate search – the search for better ways of doing things.

There are plenty of other antinomies in the world . . . .

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6 Responses to Antinomies

  1. Pedro says:

    Very good. I guess Einstein provided an answer to Kant at least.

  2. murph the surf. says:

    Funny to see a connection between physics ( the idea of time , space and matter) and economics.
    As Pedro writes while quantum mechanics has provided an explanation of Kant’s problems the related principles – Heisenbergs, Pauli’s etc all end in mind numbing explanations.
    Paired subatomic particles with spin characteristics that mirror image each other must instantaneously interact with each other if one of their spins changes across any distance ie faster than the speed of light in vacuum.How? There is no explanation I am aware of but there was an experiment which observed this phenomenon in 1997.
    Reading pages of notes of physics I’m constantly seeing symbols and formulae which look just like economics being explained.i bet there isn’t an explanation for that either!

  3. Antonios says:

    Descartes’ Meditations are a study of antinomies or the limits of reason. Of course, Descartes concluded that a God vouchsafed “clear and distinct” ideas, which Descartes defined rather arbitrarily, as definitely true. And, of course, all that God talk wasn’t at all robust, which put paid to Descartes’ Meditations.

    Hume thought much the same, but without God involved, he had nowhere to go other than a more pragmatic approach to knowledge: i.e. reason without limit is self-defeating, and pure empiricism precludes ideas such as causality and the substantial self, which makes talking about the world exceptionally difficult, so we more or less have to be careful about how we employ both sides of any knowledge claim.

    Then there’s glorious Kant and transcendental idealism, which limits the extent of our knowledge claims. Kant had basically gotten to Godel’s conclusion via philosophical means rather than mathematical.

    Anyway, I wrote an honours thesis on the limits of reason. It’s here:

    It’s here: http://anagrammatically.com/2009/04/16/the-thesis-scepticism-and-unattainable-certainty-transcendental-idealism-and-humanised-epistemology/

  4. Nicholas Gruen says:

    Thanks Antonios.

    I like your intro – and this passage:

    by even entertaining the notion that an absolute certainty can be attained, I contend that an irrefutable doubt is injected into human affairs that enfeebles other more fruitful avenues of philosophical analysis by placing too arduous a condition on its development. Such scepticism pushes enquiry further away from its anchor in the ordinary perplexities of human experience, thereby making the pursuit of certainty for its own sake the arbiter of what makes for meritable philosophy rather than the insightful analysis of the less-than-definite way we live our lives. This is not to say that pure logic, for instance, is of no consequence, or that philosophy should be a simple affair free of abstraction. Rather, philosophy needs to be aware of its own limitations as a human endeavour and engage in its enquiries accordingly if it is to best examine life and thought.

  5. paul walter says:

    I can’t begin to express the enjoyment I derived from this thread, a week after watching the last episode of Stephen Hawking’s take on creation on tel. This question has had me snookered since I came across the Parmenidians and Heraclitans of classic Greece and then the debates two thousand years later involving the likes of Galileo, Descartes, Spinoza and later between Liebniz and Newton’s proxies.

  6. Antonios says:

    One of my favourite Kant passages expresses the idea that so often comes up in Wittgenstein and has been popularised by Gladwell’s Blink, the idea that judgement requires practice. Here’s the quote:

    If understanding in general be defined as the faculty of laws or rules, the faculty of judgement may be termed the faculty of subsumption under these rules; that is, of distinguishing whether this or that does or does not stand under a given rule (casus datae legis). General logic contains no directions or precepts for the faculty of judgement, nor can it contain any such. For as it makes abstraction of all content of cognition, no duty is left for it, except that of exposing analytically the mere form of cognition in conceptions, judgements, and conclusions, and of thereby establishing formal rules for all exercise of the understanding. Now if this logic wished to give some general direction how we should subsume under these rules, that is, how we should distinguish whether this or that did or did not stand under them, this again could not be done otherwise than by means of a rule. But this rule, precisely because it is a rule, requires for itself direction from the faculty of judgement. Thus, it is evident that the understanding is capable of being instructed by rules, but that the judgement is a peculiar talent, which does not, and cannot require tuition, but only exercise. This faculty is therefore the specific quality of the so-called mother wit, the want of which no scholastic discipline can compensate.

    For although education may furnish, and, as it were, engraft upon a limited understanding rules borrowed from other minds, yet the power of employing these rules correctly must belong to the pupil himself; and no rule which we can prescribe to him with this purpose is, in the absence or deficiency of this gift of nature, secure from misuse.[28] A physician therefore, a judge or a statesman, may have in his head many admirable pathological, juridical, or political rules, in a degree that may enable him to be a profound teacher in his particular science, and yet in the application of these rules he may very possibly blunder– either because he is wanting in natural judgement (though not in understanding) and, whilst he can comprehend the general in abstracto, cannot distinguish whether a particular case in concreto ought to rank under the former; or because his faculty of judgement bas not been sufficiently exercised by examples and real practice. Indeed, the grand and only use of examples, is to sharpen the judgement. For as regards the correctness and precision of the insight of the understanding, examples are commonly injurious rather than otherwise, because, as casus in terminis they seldom adequately fulfil the conditions of the rule. Besides, they often weaken the power of our understanding to apprehend rules or laws in their universality, independently of particular circumstances of experience; and hence, accustom us to employ them more as formulae than as principles. Examples are thus the go-cart of the judgement, which he who is naturally deficient in that faculty cannot afford to dispense with.

    As far as I’m aware, although philosophers have often spoken about wisdom, no previous philosopher spoke about this very important distinction between rational knowledge and judgement so directly. At its worst, the philosophical tradition, starting from Plato, pretty much argues that reason is wisdom.

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