“Hoping for a big tent in which it is understood that disagreement is the price to be paid for exploring important ideas.”
This is conceived as an informal and spontaneous annex to my more extensive blog, Grand Strategy: The View from Oregon.
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Discord Invitation
Assimilating the World to a Logical Paradigm
The Procrustean Bed of Logic.—Since the bulk of the problem of logic is not in the process of deduction itself, but rather lies in the formalization of the problem that precedes deduction, almost any deductive framework will serve equally well. Whether we employ Aristotelian syllogisms, Stoic propositional logic, medieval terminist logic, or contemporary mathematical logic, is largely indifferent. While our modern logic is more advanced in virtue of having incorporated into itself the logical discoveries of the past twenty-five hundred years, it is not the be-all and end-all of logic. The formalization of a problem to the point at which it can be made to fit the Procrustean bed of logic is instead where we should seek the embodiment of human reason, but that would be formalization seemingly unmoored to logic, and that would deliver us over to a kind of rational vertigo—a disequilibrium of the reason, apparently without a handhold to steady ourselves. But not quite. We formalize a body of knowledge with an eye to how exactly it can be funneled into the strictures of logic, so our choice of logic for our deduction is not indifferent in this sense. The logic we employ governs the formalization we employ, so it is our choice of logic that ought to be attended by a rational vertigo, but here we have the spirit of seriousness to guide us, which, in the case of logic, is not the self-deception of ready-made values, but rather the self-deception of ready-made calculi. Given the calculus, our problem has been preemptively defined for us, as has its formalization; we have already surrendered our logical autonomy in our implied consent to the mode of formalization entailed by our chosen logic.
The Dimensions of Transcendental Logic
The Problem of Formalization.—The bulk of the problem of logic is not the process of deduction, but the formalization of the problem that precedes deduction. It is how the problem is set up for deduction rather than the deduction itself in which the art of logic lies. And here we must take the “art” of logic seriously, for that part of logic not yet reduced to mechanical derivation remains an art and not a science and cannot be mechanized in the manner of fully formalized proof procedures. How exactly we shoehorn ordinary thought into the strictures of logic so that anything at all can be derived is the real problem of logic, and this is the problem of formalization. This problem of formalization has many dimensions, all of which bear upon the result, but each of which can be taken up separately. How we break down the problem of formalization into discrete parts is itself a part of logic, or, rather, transcendental logic. Ancient axiomatics distinguished between axioms and postulates: axioms are common to all the sciences, while postulates were specific to a particular body of knowledge. We could still break down formalization in this way, but the distinction between axioms and postulates is now regarded as archaic. While the distinction remains valid, it passes over in silence specifying the choice of logic employed in derivation, unless we consider this a tacit function of the axioms. Twentieth century axiomatics distinguished between formation rules and transformation rules: formation rules govern what can be a well-formed formula (WFF), while transformation rules govern transforming one WFF into another WFF. This conceptualization has the virtue of making reasoning (transformation) explicit, but it does not explicitly thematize the elements that may be employed in a WFF (what we could call the ontology of a calculus), which is taken for granted, as the ancient conception took the univocity of reasoning for granted.
Permutations on Ernst Bloch’s “S is not yet P”
When Ernst Bloch was asked to summarize his philosophy in one line, he said, “S is not yet P.” The circumstances of this utterance were related by Harvey Cox in the Forward to Man on His Own, a collection of Bloch’s essays:
“…what does Bloch help us to see? How would his thought be capsulated if it had to be described in a few words? Bloch himself, Adolph Lowe reports, was once faced with this challenge. A few years back at a late afternoon tea in the home of a friend, someone challenged the old man to sum up his philosophy in one sentence. ‘All great philosophers have been able to reduce their thought to one sentence,’ the friend said. ‘What would your sentence be?’ Bloch puffed on his pipe for a moment and then said, ‘That’s a hard trap to get out of. If I answer, then I’m making myself out to be a great philosopher. But if I’m silent, then it will appear as though I have a great deal in mind but not much to say. But I’ll play the brash one instead of the silent one and give you this sentence: S is not yet P.” (Ernst Bloch, Man on His Own: Essays in the Philosophy of Religion, New York: Herder and Herder, 1970, p. 9)
For those who are not familiar, in textbooks of traditional philosophical logic, the syllogism was usually given the form:
All S are M.
All M are P.
Therefore, All S are P.
In such renditions of the syllogism, “S” stands for “subject term,” “M” stands for “middle term,” and “P” stands for “predicate term.” Variations on this theme yield the various “figures” and “moods” of the syllogism. (For an exposition of this kind of traditional logic cf. A. Wolf, Textbook of Logic, second edition, New York: Collier Books, 1962; however, a much more rigorous account of traditional Aristotelian logic can be found in Jan Łukaseiwicz, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, second edition, Oxford: Clarendon Press, 1972.)
Bloch gave a twentieth century twist to this ancient formulation of the syllogism. Every philosopher would have been familiar with these terms of traditional logic, so that in taking over “S” and “P” in his proposition he was appealing to something ancient in philosophy (for the ancients, the timeless embodiment of rationality that is logic), but Bloch then put the two terms into a temporal relation, and, not just a temporal relation, but a relationship of potentiality in time.
“S is not yet P” implies that S may become
P, but S and P are still, at this time, distinct, and it is by no means certain
that S will become P. It is a possibility only, one might even say that it is
an aspiration, or a hope, that S will become P, but that hope is not yet
realized; and there is also the possibility that S will not become P, or
never fully become P, but may stagnate at a point that falls short of P.
‘S in not yet P’ remains a hope only, unfulfilled. In this sense, Bloch’s
one-line summary of his philosophy is a formalization of hope, which is
eminently appropriate, as Bloch’s magnum opus is his three-volume The
Principle of Hope.
How can potentiality in time be schematized and made systematic? Can there be a logic of destiny? To schematize “S is not yet P” would involve a schematization of time, which we have in the series past-present-future, or the series before-during-after. But even the existing formalizations of time and temporal logic do not get us to the aspirational nature of Bloch’s proposition. How could a Blochean logic of hope be formulated? Would it be valid in a Blochean logic to argue:
S is not yet M;
M is not yet P;
Therefore, S is not yet P?
In other words, is hope a transitive relation? Is hope for personal salvation also hope for universal salvation, because our hope for our own fate is a transitive relation that flows on through us to others, and so on, universally? Or do we run into the fatal equivalent of ex falso quodlibet, where a false hope explosively extends itself to contaminate every authentic hope? We stand in need of a schematization that will allow us to systematically interrelate propositions such as:
“S is not yet P”
“S will become P”
“S is becoming P”
“S is almost P”
“S will never become P”
“S was once P”
“S is no longer P”
And so on. Can logic become a logic of hope, even though logic is not yet a logic of hope? Can S become P?
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Nietzsche wrote, “…the nature and degree of an individual’s sexuality reaches into the highest pinnacles of his spirit.” The converse is also true: “the nature and degree of an individual’s spirit reaches into the ultimate pinnacle of sexuality.”
I agree with both Nietzsche’s principle and its corollary, but I would add that the nature and kind of an individual’s logic – be it constructivist or non-constructivist – also reaches into the highest pinnacles of his spirit and indeed informs the world in which his spirit finds a home… or fails to find a home.
For more on the above themes see my post Nietzsche on Sexuality.
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Metaphysical Fallacies
It is possible to make a systematic, principled distinction between logical fallacies, whether formal or informal, and metaphysical fallacies? What would constitute a metaphysical fallacy?
A metaphysical fallacy would be the contravention of a metaphysical principle in reasoning. It follows from this that metaphysical fallacies are relative to a particular system of metaphysics. In other words, what constitutes a principle in one system of metaphysics will not necessarily constitute a principle in another system of metaphysics, so that the contravention of the former principle will not be a fallacy in the latter metaphysics, and vice versa.
This sounds like a fatal objection, except that exactly the same is true for logic. A logical fallacy is relative to a given system of logic. This may sound a bit odd, since if you don’t study logic you may not be aware that there are a great many systems of logic, and while many logics overlap and have principles in common, there are important differences among logics, and indeed in has been differences of principle that have driven the proliferation of logics since the formalization of logic in the early twentieth century.
Probably everyone who cares about careful reasoning is going to reject arguments directed against individuals rather than against their principles and arguments, that is to say, most will reject ad hominem arguments, but when it comes to more subtle points of reasoning, the appearance of a monolithic logic utterly collapses. These is no one logic and all hold in common, but only multiple logics and overlap and intersect.
In ordinary experience we are well familiar with the fact that other persons often have a fundamentally distinct understanding of the world than our own understanding, so that metaphysical pluralism is forced upon up, while logical pluralism is only revealed to us when we closely examine some reasoning and attempt to make the principles of its derivation explicit. As this rarely happens in ordinary experience, our logical pluralism is hidden from us in a way that our metaphysical pluralism is not.
There is not one, single metaphysics, but the many metaphysical ways of thought employed by different individuals overlap and insect. As with logic, so too with metaphysics: we might seek to define a common metaphysical core that has the consensus of all – principles of metaphysical reasoning held in common in most if not all systems of metaphysics.
W. H. Walsh, in his book Metaphysics, offers two examples of apparent metaphysical fallacies, and argues that these examples are clearly distinct from committing material or formal fallacies. The two examples he gives are
I am being driven by a friend in a motor-car when, without warning, the engine stops and the car comes to a standstill. I ask my friend what has happened: he replies that the car has stopped for no reason at all. I laugh politely at what I take to be his joke and wait for an explanation or for some activity on my friend’s part to discover what has gone wrong; he remains in his seat and neither says nor does anything more. Trying not to appear rude, I presently ask my friend whether he knows much about motor-cars, the implication being that his failure to look for the cause of the breakdown must be explained by his just not knowing how to set about the job. He takes my point at once and tells me that it is not a question of ignorance or knowledge; there just was no reason for the stoppage. Puzzled, I ask him if he means that it was a miracle, brought about by the intervention of what eighteenth-century writers called a ‘particular Providence’. Being philosophically sophisticated, he replies that to explain something as being due to an act of God is to give a reason, though not a natural reason, whereas what he said was that there was no reason for what occurred. At this point I lose my temper and tell him not to talk nonsense, for (I say) 'Things just don’t happen for no reason at all’.
…and…
A calls on B at an awkward moment when B has dropped his collar-stud and cannot find it. 'I had it in my hand a moment ago,’ he tells his friend, 'so it can’t be far off.’ The search goes on for some time without success, until A suddenly asks B what makes him think the stud is there to be found. Controlling himself, B explains that he had the stud in his hand and was trying to do up his collar when it slipped from his fingers; that there are no holes in the floor; that the windows of the room are unusually high; and that if the stud had come to pieces he must certainly have come across some bit of it after looking for so long. 'Ah,’ says A, 'but have you considered the possibility that it may have vanished without trace?’ 'Vanished without trace?’ asks B; 'do you mean turned into gaseous form, gone off like a puff of smoke or something of that sort? Collar-studs don’t do things like that.’ 'No, that isn’t what I mean,’ A assures him gravely; 'l mean literally vanished without trace, passed clean out of existence.’ Words fail B at this point, but it is clear from the look he gives his friend that he takes him either to be making an ill-timed joke or to be talking downright nonsense, a proceeding which only his being a philosopher will excuse.
I think that in these two examples Walsh was getting at what i am here calling metaphysical fallacies, and that he was attempting to do so by way of a “common core” of metaphysics shared by most, if not all, rational persons.
If we could formalize and axiomatize the metaphysics implicit in our intuitive world-views, we could point to the exact juncture at which metaphysical systems diverge, and the principles upon which they diverge, i.e., the principle that is valid in one system of metaphysics but which is not valid in another system of metaphysics.
The violation of a metaphysical principle of reasoning that belongs to the common core of metaphysics shared by almost all rational persons constitutes a more fundamental metaphysical fallacy than the violation of a metaphysical principle that is specific to a given metaphysical system and which is not to be found in other metaphysics.
In so far as a metaphysical fallacy makes use of a particular metaphysical principle the one does not oneself recognize as valid, there is still something to be learned, as a metaphysical principle that one rejects ought to have some corollary in one’s own metaphysics – either its negation or some alternative ought to be present in one’s own metaphysics, and this is likely to put one’s own metaphysical principles in a new light.
I dreamed of Gödel…
The night before last I slept a hard, dreamless sleep. I expected the same last night, but as it happened it felt as though I had had eventful, colorful dreams throughout the night (or, rather, throughout the morning). Just before I awoke, I dreamed of Kurt Gödel.
I dreamed I was at a conference at which Gödel was supposed to speak (what was that Durkheim had to say about dreaming of those who had passed away as a source for the idea of the immortality of the soul?) but the lectern was empty. I called a phone number that I had and was told that he wouldn’t be appearing after all, but that I could come to visit.
I went to a dingy apartment and a woman answered the door. She let me in and I began to ask her about Gödel and she simply pointed to him sitting down further inside the apartment. I went over to him and began speaking, first just with a few pleasantries. He didn’t respond, so I kept talking. I began thinking that maybe he couldn’t hear me or maybe he was no longer responsive.
Then he suddenly began to speak. He spoke rapidly but in a clear and coherent voice, though I couldn’t follow most of what he said. He took my notebook out of my hands and flipped through it, and then said the one thing I heard clearly and can remember: “You’re three hundred years out of date.”
Unlike his photographs, the Gödel of my dreams had a large beard, not unlike Santa Claus.
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P or not-P
The past few days has seen an interesting discussion emerge on the Foundations of Mathematics listserv (FOM), which grew out of link to a recent article by Frank Quinn in Notices of the AMS, “A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today.”
Quinn’s article highlighted the role of the Law of the Excluded Middle (also known as tertium non datur) in mathematical reasoning. The Law of the Excluded Middle is a logical law that is usually stated, “P or not-P.” Though this sounds simple enough, it has been a sore spot in mathematics because it allows one to “prove” the existence of something that one can neither construct or exhibit – once all the alternatives are eliminated (and with classical negation, there is only one alternative), one is left with the mathematical equivalent of the last man standing.
Dutch philosopher of mathematics L. E. J. Brouwer formulated one of the most influential schools of constructivist thought, intuitionism, by jettisoning the Law of the Excluded Middle and simply doing without it.
Frank Waaldijk (like Brouwer, from the Netherlands) wrote on the FOM list:
The revolution in mathematics spreads much further than `just´ methodology. The revolution is about the concepts underlying all of our thinking about math, science and reality.
And my preliminary conclusion is: we still know nothing for sure. We are stumbling in the dark.I therefore cannot take Quinn’s stance on the role of `excluded middle´ in mathematics very seriously. Classical mathematics, in its full-fledged embrace of excluded middle, can be compared to science fiction…or dreamland if you would like a stronger metaphor. It’s nice to dream, and nice to be able to conjure battlestars and time travel and black hole mining and…But it is also important to return to reality from time to time. This is where constructive mathematics comes in. Constructive mathematics and classical mathematics are not always at odds per se…it is `just’ a major difference of focus and perspective. But I am personally convinced that we need constructive mathematics for a better understanding of our physical world and physical reality. And constructive views on excluded middle should already be taught in high school, not exclusively but at least for comparison.
Panu Raatikainen responded to Frank Waaldijk writing:
These are strong claims, and we’ve heard them now and then before, but it would be nice to hear some convincing arguments supporting them… I’ve honestly tried hard to find one for some time, but have so far failed…
Today I wrote the following (though with additional material added below) to the FOM list:
Perhaps a more charitable conception of the relationship between classical eclecticism and its tolerance of non-constructive modes of reasoning on the one hand, and on the other hand the many species of constructivism that have been proposed in order to place limits on classical eclecticism, is to be found in an image proposed by the mathematician Alain Connes:
“Constructivism may be compared to mountain climbers who proudly scale a peak with their bare hands, and formalists to climbers who permit themselves the luxury of hiring a helicopter to fly over the summit.”
(Conversations on Mind, Matter, and Mathematics, Changeux and Connes, Princeton, 1995, p. 42)
On the next page Connes says, continuing the image,
“…the uncountable axiom of choice gives an aerial view of mathematical reality – inevitably, therefore, a simplified view.”
If we think of the constructivist perspective very roughly as a “bottom up” approach, like a mountain climber who starts at the base and clambers over every cliff and every ledge on the way up, and non-constructive methods as a “top down” approach, an aerial view of mathematics, perhaps lacking in definite detail, but giving the big picture of the scene, then the two approaches are complementary.
An adequate conception of mathematical reality must include both constructive and non-constructive approaches, rather than dismiss classical mathematics as science fiction or dreamland.
I suggest that the top-down perspective of classical mathematics and the bottom-up perspective of constructivism meet in the middle, and that middle is constituted by macroscopic mathematical intuitions – the familiar instances of mathematical knowledge and experience like counting with cardinal numbers.
The classical foundationalist project plunges down from the heights and seeks to immerse itself in the details of mathematical knowledge from above. The relationship of the foundationalist to foundations established regressively (in Russell’s sense) from macroscopic mathematical intuitions is analogous the relationship believed by the classical mathematician to hold between macroscopic mathematical intuitions and the mathematical reality from which they are thought to descend. Thus the foundationalist project is a project to bring down the truth of macroscopic mathematics down its foundations.
The constructivist is no believer in the truth or truth-giving properties of macroscopic mathematics, which he regards as riddled with errors. The constructivist seeks to build from below only what can be built step-by-step, content to neglect the big picture and therefore blind to the landscape in which he patiently builds. To the constructivist, and foundationalist is falling off a cliff when he plunges downward; to the classical mathematician, the constructivist is so absorbed in this life on firm ground, with his feet in the mud, that he has ceased to dream and no longer looks up at the stars.
To put the distinction between the two in a quasi-scientific idiom, constructivism “explains” macroscopic mathematical intuitions as being constructed from ur-intuitions (as, for example, from Brouwer’s first and second acts of intuitionism), while classical eclecticism “explains” macroscopic mathematics intuitions from the top down, with reference to the abstract mathematical entities, from which flow macroscopic mathematical intuitions when the mind directly “perceives” mathematical reality (as in Gödelian mathematical realism, where the mind possesses a special faculty for “perceiving” mathematical reality).
The constructivist, who is a mathematical fox and knows many little things, many details of mathematics, and the classical mathematician, who is tolerant of non-constructive methods and as a mathematical hedgehog knows one big thing, need each other.
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Sources of Conceptual Confusion
Conceptual confusion can be the result of committing a logical fallacy, in which case one is confused because one is making an error in one’s reasoning without realizing that one is reasoning fallaciously.
However, conceptual confusion can also come about as a result of attempting to observe a logically valid distinction that is not yet widely recognized (i.e., a distinction that is so commonly conflated that it is not recognized as a conflation) and which is therefore difficult to formulate within a known logical framework.
When a distinction is intuitively felt but not yet made fully explicit, one is likely to swing back and forth between making the distinction where it stands out clearly, even if not explicitly formulated, and conflating the distinction where it appears in a less intuitively obvious context.
In this latter case, one is attempting to reason validly, but the linguistic and conceptual infrastructure ready-to-hand does not have the resources to accommodate the distinction toward which one is groping.
One can experience a certain cognitive dissonance as a result of attempting to reason validly but being channeled into something like a fallacy (or, at least, a conflation) by the received logical theory one is employing.
Thus one can experience conceptual confusion either as a result of cognitive inadequacy (personal error) or logical inadequacy (systemic error).
