Re: R.J. Lipton and K.W. Regan • Isomorphism Is Where It’s At
Just off the top of my head, a couple of examples come to mind.
Sign Relations. In computational settings, a sign relation 
Cactus Graphs. In particular, a variant of cactus graphs known (by me, anyway) as painted and rooted cacti (PARCs) affords us with a very efficient graphical syntax for propositional calculus.
I’ll post a few links in the next couple of comments.
Re: Cristopher Moore on Theorems From Physics?
It is critically important to distinguish between the objective landscape, the boolean functions as mathematical objects, and the syntactic landscape, the particular formal language we are using as a propositional calculus to denote and compute with those objects. If we do hill-climbing, we must keep our feet on the objective territory, however much we rely on syntactic maps to narrate the travelogue. (Many will be thinking of manifolds here.) The object domain has a fixed structure but the conceptual clarity and computational efficiency of propositional calculi can very likely be improved indefinitely.
Re: R.J. Lipton and K.W. Regan • Theorems From Physics?
Bits of Synchronicity …
What kind of information process is scientific inquiry?
What kinds of information process are involved in the various types of inference — abductive, deductive, inductive — that go to make up scientific inquiry?
What kinds of information process are computation and proof?
Many types of deductive inference, including many kinds of computation and proof, don’t really change our state of information so much as increase the clarity of that information. Do we have any way to quantify clarity in the way we define measures of information?
It’s been a while since I threaded this thread — and then there were all the delightful distractions of the holiday convergence — so let me refresh my memory as to what drew me back to these environs.
I’m still in the middle of trying to catch up on some long put-off work, but recent discussions of logical graphs and physics and the like on the list have bestirred me from my grindstone long enough to pass on a few links to the things I’ve been doing along those lines. This is all “Alpha” as far as Peirce’s graphology goes, but one of the things we’ve learned in recent decades from computational complexity theory is just how key a role problems like propositional calculus play in solving many other problems of practical interest, so I won’t make any further apology for focusing attention on this “zeroth order” level. I don’t have much to say about physics per se but if we generalize our concept of dynamics and speak of systems theory as a study of media and populations that move through their state spaces over spans of time, then I think it is useful to take up that perspective on the time evolution of logical media informed by logical signs.
Good — logic and time, the time evolution of inquiry driven systems. I’ll do my best to stay focused on that interplay of subjects.
Let me start with a different set of articles this time. You will notice a lot of redundancy among these articles, as I’ve written many invitations to the subject over the years. Some of these were originally written for people who were already familiar with Spencer Brown’s Laws of Form, so I jumped right in with the formal equations that would have been recognizable to them.
I need to stay with this problem a while …
In contemplating this problem I always find it helpful to ruminate on the diagram shown above — I might even call it a mandala for its wealth of symbolic features and its aid in organizing the pro-&-con-fusion of mental impressions.
Here is the corresponding text from Aristotle and the context that leads on to Peirce:
Re: Peirce List Discussion • Tom Gollier
Here my task is to build bridges between several different classical and contemporary uses of the word model, so I don’t have the luxury of complete control over the words in play but have to start from the customary senses in the various communities of interpretation. Of course I’m slyly working from a sign-relational backdrop, but I have to be sleight-handed about that and not hit people over the head with it.
You can probably guess that I’m using object to cover sign-relational objects, and theories are clearly syntacked together from complexes of sign-relational signs, so all we have left to pin down is where the various kinds of model sit at the table set with the labels of Object, Sign, Interpretant.
In its theoretical sense, a model of a theory is anything the theory is true of, anything that satisfies the theory. In that sense, a model is very like an object. It is whatever the theory is talking about. In the order of nature, indeed, models come before theories. But there is another order, the order of art, and one may construct artificial models out of almost any stuff, even the stuff of signs. So you see the kind of wiggle room we have to work with.
Things are easier outside of logic, in applied mathematics and the special sciences, where models are just things like analogues, icons, simulations, and similar representations of objects. But that makes them objects serving as signs of other objects, and so you may find some semiotic subtlety lurking there.
Re: K.W. Regan • The Graph Of Math
Re: Artem Kaznatcheev • Three Types Of Mathematical Models
What — if anything — is the common sense that connects the different senses of the word model, as it has been used over the years in logic, mathematics, and the special sciences? It’s a problem I’ve been running into for several decades now and I think I can trace the roots of it going back as far as Aristotle’s treatment of analogy. Just by way of creating a bit of trans-disciple-ary interaction, here’s a link to a link of the issue’s most recent arising in my own roll of blogs.
I am still in the middle of trying to catch up on some long put-off work but recent discussions of logical graphs and physics and the like on the Peirce List have bestirred me from my grindstone long enough to pass on a few links to the things I’ve been doing along those lines. This is all “Alpha” as far as Peirce’s graphology goes but one of the things we’ve learned in recent decades from computational complexity theory is just how key a role problems like propositional calculus play in solving many other problems of practical interest, so I won’t make any further apology for focusing attention on this “zeroth order” level of logic. I don’t have much to say about physics per se, but if we generalize our concept of dynamics and speak of systems theory as a study of media and populations that move through their state spaces over spans of time, then I think it is useful to take up that perspective on the time evolution of logical media informed by logical signs.
Please remember that we are not now concerned with the physics and chemistry, the anatomy and physiology, of man. They are my daily business. They do not contribute to the logic of our problem. Despite Ramon Lull’s combinatorial analysis of logic and all of his followers, including Leibniz with his universal characteristic and his persistent effort to build logical computing machines, from the death of William of Ockham logic decayed. There were, of course, teachers of logic. The forms of the syllogism and the logic of classes were taught, and we shall use some of their devices, but there was a general recognition of their inadequacy to the problems in hand. […] The difficulty is that they had no knowledge of the logic of relations, and almost none of the logic of propositions. These logics really began in the latter part of the last century with Charles Peirce as their great pioneer. As with most pioneers, many of the trails he blazed were not followed for a score of years. For example, he discovered the amphecks — that is, “not both … and …” and “neither … nor …”, which Sheffer rediscovered and are called by his name for them, “stroke functions”.
It was Peirce who broke the ice with his logic of relatives, from which springs the pitiful beginnings of our logic of relations of two and more than two arguments. So completely had the traditional Aristotelian logic been lost that Peirce remarks that when he wrote the Century Dictionary he was so confused concerning abduction, or apagoge, and induction that he wrote nonsense. Thus Aristotelian logic, like the skeleton of Tom Paine, was lost to us from the world it had engendered. Peirce had to go back to Duns Scotus to start again the realistic logic of science. Pragmatism took hold, despite its misinterpretation by William James. The world was ripe for it. Frege, Peano, Whitehead, Russell, Wittgenstein, followed by a host of lesser lights, but sparked by many a strange character like Schroeder, Sheffer, Gödel, and company, gave us a working logic of propositions. By the time I had sunk my teeth into these questions, the Polish school was well on its way to glory.
In 1923 I gave up the attempt to write a logic of transitive verbs and began to see what I could do with the logic of propositions. My object, as a psychologist, was to invent a kind of least psychic event, or “psychon”, that would have the following properties: First, it was to be so simple an event that it either happened or else it did not happen. Second, it was to happen only if its bound cause had happened — shades of Duns Scotus! — that is, it was to imply its temporal antecedent. Third, it was to propose this to subsequent psychons. Fourth, these were to be compounded to produce the equivalents of more complicated propositions concerning their antecedents.
In 1929 it dawned on me that these events might be regarded as the all-or-none impulses of neurons, combined by convergence upon the next neuron to yield complexes of propositional events. (McCulloch 1965, 7–9).
144. But since you propose to study logic, you have more or less faith in reasoning, as affording knowledge of the truth. Now reasoning is a very different thing indeed from the percept, or even from perceptual facts. For reasoning is essentially a voluntary act, over which we exercise control. If it were not so, logic would be of no use at all. For logic is, in the main, criticism of reasoning as good or bad. Now it is idle so to criticize an operation which is beyond all control, correction, or improvement.
145. You have, therefore, to inquire, first, in what sense you have any faith in reasoning, seeing that its conclusions cannot in the least resemble the percepts, upon which alone implicit reliance is warranted. Conclusions of reasoning can little resemble even the perceptual facts. For besides being involuntary, these latter are strictly memories of what has taken place in the recent past, while all conclusions of reasoning partake of the general nature of expectations of the future. What two things can be more disparate than a memory and an expectation?
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147. The second branch of the question, when you have decided in what your faith in reasoning consists, will inquire just what it is that justifies that faith. The simulation of doubt about things indubitable or not really doubted is no more wholesome than is any other humbug; yet the precise specification of the evidence for an undoubted truth often in logic throws a brilliant light in one direction or in another, now pointing to a corrected formulation of the proposition, now to a better comprehension of its relations to other truths, again to some valuable distinctions, etc.
148. As to the former branch of this question, it will be found upon consideration that it is precisely the analogy of an inferential conclusion to an expectation which furnishes the key to the matter. An expectation is a habit of imagining. A habit is not an affection of consciousness; it is a general law of action, such that on a certain general kind of occasion a man will be more or less apt to act in a certain general way. An imagination is an affection of consciousness which can be directly compared with a percept in some special feature, and be pronounced to accord or disaccord with it. Suppose for example that I slip a cent into a slot, and expect on pulling a knob to see a little cake of chocolate appear. My expectation consists in, or at least involves, such a habit that when I think of pulling the knob, I imagine I see a chocolate coming into view. When the perceptual chocolate comes into view, my imagination of it is a feeling of such a nature that the percept can be compared with it as to size, shape, the nature of the wrapper, the color, taste, flavor, hardness and grain of what is within. Of course, every expectation is a matter of inference. What an inference is we shall soon see more exactly than we need just now to consider. For our present purpose it is sufficient to say that the inferential process involves the formation of a habit. For it produces a belief, or opinion; and a genuine belief, or opinion, is something on which a man is prepared to act, and is therefore, in a general sense, a habit. A belief need not be conscious.
Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Volume 2 : Elements of Logic, 1932. Reprinted with corrections, 1960.
Inquiry Into Inquiry 