Philosophy

The logic of the human mind

I am particularly concerned with human logic and to explaining how it is different from the logic of machines.  I regard it as a matter of great urgency that we should clarify the distinction between the mind and digital machines.
    Already in my undergraduate days as a student of Philosophy at Cambridge, I was led to study mathematical logic and the philosophy of mathematics with a view to uncovering the answer to the distinction between the mind and digital computers.  I recognised that such a project would require a great deal more mathematical knowledge then I was then equipped with.  On leaving Cambridge I became a teacher of English literature, but nonetheless started to study Mathematics graduating with the highest first, and subsequently completing a Masters Degree.  It was my intention at the time to continue to study for a PhD, but career problems forced me to open my own school, and I subsequently lost fifteen years of continuous study of logic.  On closing my school I decided to devote time once again to an intellectual puzzle that I regarded as of immense importance.
    Here I present the results of my studies.   Historically, my first paper was Poincare's Thesis.  Essentially, the great mathematician Poincare had already solved the problem of the distinction between human and machine logic by pointing out:

"mathematical reasoning has of itself a kind of creative virtue, and is therefore to be distinguished from the syllogism. "
                                                                Henri Poincare: Science and Method

    In a series of essays in his Science and Method Poincare demonstrated that the distinctive process of reasoning called mathematical induction could not be reduced to the analytical reasoning of machines.  He also lampooned "the art of writing a treatise on mathematics without using a single word of ordinary language”.  His warnings were ignored by a mathematical community that would subsequently narrow the nature of mathematical reasoning so that it would become identical with machine logic.

A lattice

    My task was to explain in detail and with due regard to the extraordinary superstructure of modern first-order mathematical logic just why Poincare was right.  In essence I discovered a criterion for what limits machine logic - namely, that it is constrained by an underlying structure called a lattice.  The lattices in question are made up of discrete points.  On analysis of Godel's proof of the incompleteness of mathematics I found that an extended version of his theorem that could not be a point of any such lattice.  So human logic is transcendent over machine logic.

    I also revisited a problem that I had studied at Cambridge.  It has long been known that there are processes that a machine cannot compute.  One such problem is known as the Halting Problem.  Proponents of Artificial Intelligence argue that human logic is constrained in exactly the same way as machines - and although a digital machine cannot solve the Halting Problem, neither can a human mind.  But if a solution to the Halting Problem could be found, then it would categorically refute the claims of strong AI.  I believe I have found just such a solution, which, not surprisingly, rests on a simple inductive argument, and thus confirms Poincare's original thesis.
    While studying Poincare's thesis I started investigating transcendental numbers and the structure of the continuum.  I discovered that transcendental numbers are specifically constructed as generic sequences.  This in turn led me to investigate the structure of the continuum, which is the mathematical description of space.  There is a very deep unsolved problem concerning the continuum, namely the Continuum Hypothesis, dealing with the question of how many points there are in space.  I realised that the theory of transcendental numbers rests on the Axiom of Completeness and that it was possible to solve this problem of the continuum by simply exploring the logical consequences of that Axiom with particular regard to the Mahler classification of transcendental numbers.  It is an extraordinary observation that there appears to be not one single attempt to use this strategy in the whole of the literature, which is vast and very complicated.
    That omission on the part of the mathematical community requires some explanation.  One view is that over the past century there has grown up a false belief that all mathematics is first-order set theory.  Therefore, the whole investigation was constrained by the assumption that only first-order set theory can be considered.  Now the Axiom of Completeness is specifically second order.  So we have here just another confirmation of what the difference between human logic and machine logic is: machine logic is first-order, human logic is second-order.
    The theory of transcendental numbers was developed independently from set theory in the pre-war period, and it seems that there was no cross-over between this discipline and that of set-theory.
    Set theory is founded on a single "primitive" relationship - that of set membership.  It is intuitively self-evident that the claim to be able to explain everything in science by a single relation of this type is absurd.  My investigation into the continuum also showed that there is a second primitive relation, that of extension.  We cannot derive that an object is extended in space from the mere fact that it is the member of some set.  It is also because this primitive quality of space has been ignored by set theorists that they have been unable to resolve the continuum problem.
    If there is a single relation that cannot be reduced to set membership, then again, the human mind is not a digital computer.  One such relation is extension.  Fundamentally, the human mind sees and a computer processes: digital machines have no phenomenological experience from which to derive concepts; they lack all perceptual concepts whatsoever.  Among these perceptual concepts is that of extension.
    It has been a characteristic of contemporary philosophy that anyone objecting to the view that digital machines are minds gets quickly shot down by the philosophical community. For this reason I wish to make a few more observations about my general position.

1.    Firstly, more than anything I am dedicated to the truth, and to the dialectic method as a means to finding the truth.  I would be delighted to hear from anyone who could demonstrate a weakness, fatal or otherwise, to the project I advance here.   That is also the only basis on which I would modify, withdraw or retract any of the claims made in my papers.  I also point out that I use no argument previously brought to light and criticised by other authors.  It's all original. (* But see footnote.)
2.    Secondly, I am an advocate of the empirical method.  The philosophy that I am objecting to has a strong tendency to turn science into a game or convention: truth by definition or by coherence.  But the scientific method requires that truth be established by evidence, and this makes it a synthetic project of the human mind to understand its environment.  So I am a champion of science, one who resoundingly rejects the view that everything that we can know can be put into a box labelled "set membership".
3.    Thirdly, I am by no means rejecting the scientific study of the mind, which like everyone I find to be fascinating and full of surprises.  But the human mind is not a first-order system.  Curiously, the only first-order systems that have any physical existence are those that we artificially construct - digital computers.  Organic chemistry is not first-order; the brain is not first-order.  So let us, by all means, have the second-order science of the mind - love it and embrace it!  You do not have to be a first-order set theorist to scientifically study the mind.  The brain is a machine - true - but a second-order machine.
4.    Fourthly, I have no particular religious affiliation.  I am not trying to convert anyone to some sect.  Of course, and only for the record, I feel hugely indebted to Christian ethics, and like to think I base my practical actions up0n it.  I also have a high regard for all the major world religions.  But this has nothing to do with whether the human mind is or is not a digital computer.  And indeed, whatever my motives might be, they could not be objections to the truth of what I assert.  Only objective argument against the statements asserted in these papers could undermine them.

* I am deeply indebted to the work of J. R. Lucas, whose paper Minds, Machines and Godel is seminal.  His work makes extensive implicit use of Poincare's thesis.  Lucas also makes the point that mathematical reasoning is at least second-order.  It turns out that the claim that Lucas cites in his Satan Stultified essay, which he attributes to Pual Benacerraf, "that the argument is not, and cannot be, a purely mathematical one" is not quite true.  Although this is a debate about the foundations of mathematics, and hence cannot ever be "purely mathematical", I provide extensive, rigorous mathematical treatment of the problem and also formalise the following statement of Lucas: "[Godel's theorem] ... can be taken as a formal proof sequence yielding certain syntactical results about a certain class of formal systems, but it can also be taken as giving us a certain type or style of argument, which we can understand, and, once having got the hang of it, adapt and apply in innumerable different circumstances."  This is an induction in the metalanguage and corresponds to the construction of an ultrafilter transcendent over any locally compact Boolean lattice.  For the details, please see my work on Poincare's thesis. 

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I started studying the Cantor set for the purpose of clarifying the underlying structure of machine logic, so as to compare it with the inductive arguments we find in mathematics generally, and the specific one I uncovered in Godel's theorem.  On the Continuum work builds on that foundation and is a clearer description of that structure.  Also, there are a series of Flash presentations, which provide an introduction to the whole paper.
    This paper has a supplement, Beyond the Completeness Axiom. 

Beyond the Axiom of Completeness consolidates the results of the first paper by making an initial exploration of alternative models of the continuum; thus showing that the model provided by the Axiom of Completeness is in some sense the "simplest" one.

My original paper, Poincare's Thesis, is an amalgamation of themes.  At present, the "plot" of the book weaves through three themes:
1. The investigation of the logical foundation of digital processes and the demonstration that a generalised version Godel's incompleteness theorem cannot be embedded in this structure.  Hence, the transcendence of human logic over machine logic.  I am aware that Godel's theorem in its usual formulation is a first-order theory; the argument I present is not an obvious failure to meet that objection.
2. The solution to the Halting problem.  I provide an algorithm for this solution, and also explain why the human mind can solve the problem, whereas computers cannot.  In a subsequent section I build on this work to analyse Lagrange's theorem, which has resisted the attempts of computer theorists to provide a machine logic variant.  I show that Lagrange's Theorem is intimately connected to the Halting problem and so explain why there will never be a formalisation of Lagrange's Theorem.
3. I develop an alternative Neo-Kantian philosophy of mathematics, a counterbalance to the prevalent formalism of our times.  I seek to initiate the exciting investigation into what human logic really looks like, building on Leibniz's universal characteristic.  I provide alternative solutions to all the major paradoxes of logic, including the liar paradox.

DOWNLOADS

On the Continuum
Beyond the Continuum
Poincaré's Thesis

Introduction to Poincare's Thesis (Flash presentation)
This introduction is non-technical.

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