In 1973, the British philosopher Michael Dummett was invited to deliver the annual philosophical lecture at the British Academy. He took as his subject ‘The Justification of Deduction’ – how to justify the ordinary practices of mathematicians
if they are useful, what more justification is required? Math, like other disciplines, needs resources. That which is useful, makes money. Math progresses fastest when it makes money by raising producitivity.
without helping oneself to the assumption that their subject matter exists in a realm of its own.
Whether this is Plato's realm of forms or South Park's 'Imagination-Land' scarcely matters. The only question is, 'can the thing pay for itself'.
The problem, he told his audience, was perhaps ‘one of the most fundamental and intractable problems … in all philosophy’.
Philosophy is merely a displacement activity. Dummett, it must be said, introduced low IQ peeps like me to one or two elementary mathematical ideas.
What Dummett took from Frege’s work on logic was that the problems of mathematics are, at heart, problems of language.
The foundations of mathematics are not necessarily what we would call mathematical. On the other hand, the foundations of those foundations may well be mathematical because, in the ultimate analysis, everything is.
Where things had gone wrong, he thought, was that Frege had grounded his account in a mind-independent notion of truth.
All protocol bound financial, legal, or scientific accounts have that feature. On the other hand, my account of my life as a prostitute in Hollywood in the Eighties is not grounded on anything save the mind-dependent notion that I'd look utterly ridiculous playing the Julia Roberts role in @prtty Woman'.
‘In this respect (and in this respect alone),’ Dummett wrote, ‘Frege’s new approach to logic was retrograde.’
Why posit mind-dependent entitites? For all we know, we are meat machines merely. More advanced species truly have minds. The proof is, their minds can change reality at the most fundamental level.
For Frege, logic answered to truth: the rules were good because they preserved something already fixed.
only if there was 'restricted comprehension'. Otherwise, you got nonsense
Dummett turned this around.
Following Wittgenstein’s later thinking, Dummett argued ‘a theory of meaning is a theory of understanding.’
is a theory of information storage, retrieval, or exchange which in turn is an economic theory of a trivial type.
And understanding, he insisted, must be manifestable: whatever a competent speaker knows in knowing what a sentence means must be something she can, in principle, display in her use of language.
Suppose there is a sentence you hear on the point of death which explains to you the central mysterey of your life. It can be uttered at no other moment while conveying the same meaning. Can it, in principle, be conveyed? Not if it causes death unless our 'principle' stipulates for an after-life. But that may contradict the meaning received or, rather, prevent its reception in the intended form.
More generally, unless meaning is understanding and both remain fixed, some problem of the above type arises, at least in principle, under any and every imaginable set of circumstances.
In The Logical Basis of Metaphysics (1991), Dummett gave this account of language, logic and reality its most sustained form. Our linguistic practices – the ways in which we are prepared to assert, justify and infer – determine logic.
Only if that is what we habitually do and even then, this would only be true for ourselves.
Logic, in turn, determines what counts as a fact,
Sherlock Holmes may use logic to show us that what we accept as fact can't be any such thing. But this is a story about a guy who is smarter than us determining what we come to believe.
what counts as truth, and ultimately what we are entitled to regard as real.
Sadly, for most of us, earning a living determines what counts as truth. Also, I'm not really the star of 'Pretty woman' even though many people mistake me for Julia Roberts. Sadly, it appears she has gained a lot of weight. That's why I am often referred to as 'fat cunt'.
Consider how a child learns the word ‘red’.
The same way she learns the name of things she sees
We do not begin by telling her about light of a certain wavelength, or the metaphysics of colour properties, or the conditions that would make ‘this is red’ true. We point at things.
Or she points at things and we say 'do you want the red one or the blue one'?
We correct her when she calls a ripe tomato green; we approve when she calls a fire engine red. Over time, she comes to use the word as we do – and at that point, she understands it.
A child may indeed be said to understand God as well as we do. But it is unlikely to understand the nature of Diplomacy even if its daddy is a diplomat.
There is no further fact, hidden behind the practice, in which her understanding consists. The understanding is the practice, competently exercised.
The practice can dispense with the understanding. If you are giving your g.f plenty of orgasms, she tends to feel that there's something important about her which you understand very well. If this isn't the case, it may be a good idea to attend one of those courses on communicating with your partner during sex and understanding each other's needs & (let's face it) limitations.
Dummett died in 2011, having been knighted in 1999 for ‘services to philosophy and to racial justice’.
he was a convert to Catholicism. At that time, some Christian Churches, believed in the natural inferiority of black people.
He is remembered as one of the most significant British philosophers of the previous century. Yet for all its power, his programme remained a prospectus: he set out what a theory of language, logic and reality must achieve, but he never completed one.
But he was less shite than others of his ilk.
In 2005, Tor Sandqvist, a doctoral student at Uppsala University, submitted a thesis
He showed first-order classical logic could be validated using solely constructively valid reasoning in the metalanguage, eliminating the reliance on a priori "bivalence" (the strict assumption that everything is fundamentally True or False)
that took up Dummett’s challenge: to justify the ordinary practices of mathematicians without any appeal to truth.
His base-extension semantics provides mathematical accounts for intuitionistic and classical systems based entirely on inductive rules of inference rather than traditional model theory.
Where Dummett had given the conditions a solution must satisfy, Sandqvist gave a solution. He produced a precise mathematical justification of deduction as mathematicians actually use it, grounded entirely in practices of assertion. The boxer’s fists were untied.
Now it appears, the boxer may be replaced by an AI. Sandqvist & Voevodsky appeared together at Conferences till the latter passed away in 2017.
. Gheorghiu reframes incompleteness as a divergence between two internally determined notions of consequence associated with a single theory, rather than as a gap between syntactic provability and truth in a mind-independent structure. The result clarifies the relationship between reflection, consistency, and inferentialist approaches to meaning, and shows how substantial semantic determinacy may arise from the inferential structure of arithmetic itself.
It was rediscovered a decade later in the Department of Computer Science at University College London, where a group of logicians recognised what Sandqvist had actually built, and developed it into a full framework on a par with the orthodox truth-conditional view. The framework now stands equal to the truth-based account, analogous to the situation after Gödel’s PhD thesis. What about mathematics? The UCL team have recently shown that the Dummett-Sandqvist framework gives a way to understand what Gödel’s Incompleteness Theorems are about, and it is not the limits of knowledge.
If meaning comes from practices of assertion, then our concepts are settled only as far as our practices have settled them.
In other words, the thing is a moveable feast. But focal solutions for coordination games cshift all the time.
A system of definitions is a kind of constitution that determines something. The US Constitution, for example, is meant to enshrine a form of democracy.
It was meant to solve the problem created by dual sovereignty. Southern Democrats abandoned a racist 'form of democracy' when I was about 2 years old.
But what a constitution defines answers to how it is amended.
It may do. It may not. The Warren Court was described as having brought about a 'Constitutional Revolution'. Some say that Trump appointed Judges are bringing about a Counter-Revolution.
Principia Mathematica is a constitution for arithmetic.
without algorithms for things like long division
To read it as a constitution is to assume it is consistent; otherwise, it would be nonsense.
We only need to assume that there could be a Judge Hercules giving it a 'harmonious construction'.
So, when we amend it by asserting that ‘Principia Mathematica is consistent,’ we are not changing the subject: the new axiom says aloud only what the practice already did. Gödel’s theorems apply afresh to the enlarged constitution: the amended system has its own unprovable truth, its own consistency it cannot certify.
The US constitution is 'buck stopped' in that the Supreme Court has the final say unless 'doctrine of political question' arises. But this is a matter of economia, not akriebia.
No amendment is the last; each faithful extension opens room for the next. To grasp a concept like arithmetic is not to survey a finished structure;
the 'extension' is epistemic, impredicative & not well defined.
it is to hold a principle for going on: a rule for laying the next stretch of line, not a map of where the line ends.
Thus we don't have to be silent about anything of which we can't speak precisely.
Dummett called this openness indefinite extensibility.
There could be (using Kripke schema) an end to mathematical time when all choice sequences become 'determinate'.
Faithful is the word doing the work, because not every amendment is. Gödel claimed the US Constitution could deliver a dictatorship.
This was obvious at the time. That's why Presidential terms were restricted. Otherwise the Executive could pack the Bench & then go in for gerrymandering.
That does not make dictatorship a form of democracy.
Why not? The Weimar Constitution explicitly provided for a Caesarean President. In practice, the US has this. Mutually Assured Destruction means that POTUS could pull the plug on all life on earth. That's pretty fucking dictatorial.
Whatever loophole he had in mind, it would have contradicted the practices the Constitution already enshrined. An amendment can betray what a constitution defines. But it can also unfold it, drawing out what the founding commitments demanded all along.
White heterosexual men who owned slaves never intended for the American Constitution to protect gays, blacks, First Nations & women.
Only the second kind is what Dummett has in mind.
Dummett & Farquarhson (who was White South African) hoped that some ideal type of constitution could be created to safeguard minorities. Sadly, the thing was a pipe dream.
The constitution of our language is our current practice for public discourse.
Sadly, public discourse is stuff Trump says, not what erudite savants talk about.
As it is extensible in this way, what we claim as ‘true’ is not a fixed structure with a ceiling we are forever failing to reach, but something that can change. Gödel’s verdict never applied to mathematics, but only to some edition of the constitution. Within the Sandqvist system, we can attend to the extension systematically. Hilbert’s words outlive his programme: Wir werden wissen. We shall know. Not because the answers are waiting at the end of a ladder, but because truth goes exactly as far as we have built the rungs.For Per Martin Lof knowing a proposition means explicitly possessing a constructive proof or verification. In that sense, AIs may soon know math better than most early career mathematicians. Even as we speak, more rungs may being added such that, as Terence Tao says, AI 'jumping robots' can by pass the slow collobarative process of mathematicians assisting each other to find hand holds and footholds to ascend further up the mountain.
No comments:
Post a Comment