Permanent relationships are described in an impermanent idiom, which reflects both the age’s climate of thought and the individual philosopher’s personal style of thinking. No philosopher understands his predecessors until he has re-thought their thought in his own contemporary terms; and it is characteristic of the very greatest philosophers, like Kant and Aristotle, that they, more than any others, repay this effort of re-thinking.
-P.F. Strawson, Individuals
I have something of a fetish for bound variables.
Frege repeatedly makes the claim that when one predicates truth of a thought, one does not say any more than is said in the expression of the original thought. In “On Sense and Nominatum,” Frege writes, “One could virtually say: `the proposition that 5 is a prime number is true.’ But on closer examination one notices that this does not say any more than is said in the simple sentence `5 is a prime number’.” And in “Thoughts,” he writes, “nothing is added to the thought by my ascribing to it the property of truth.”
Here is a natural way to interpret Frege’s view. Truth appears to be unique in that when it is predicated of a thought, we do not wind up with a different thought. This seems to be an identity claim: “A thought P is identical to the thought that P is true.” Or, take T to be the truth-predicate. Then, for any thought P, the thought T(P) is identical to P. This is, moreover, a stronger claim than that “A thought P is logically equivalent to the thought that P is true.” It is a stronger claim because one might wish to hold that two distinct (so, non-identical) thoughts can be logically equivalent. Additionally, the claim that P and T(P) are logically equivalent amounts to a claim that they cannot fail to have the same truth-value. Frege takes truth-values to be the nominata of thoughts. But the identity claim we are considering says more than that P and T(P) cannot fail to have the same nominatum. It says that the thoughts themselves are identical.
But one might be skeptical of this identity claim. Consider the following example: At dinner, John says “Paris is in France.” This sentence expresses a certain thought––call it “F.” Now, if I say “It is true that Paris is in France,” then I use the expression “Paris is in France” to refer to the thought F, and I ascribe the property of truth to this thought. So, according to the above identity claim, the thought expressed by “It is true that Paris is in France” is identical to the thought expressed by “Paris is in France.” But now suppose that I say “What John said at dinner is true.” Here, the expression “What John said at dinner” refers to the thought expressed by “Paris is in France”––it refers to F. And, once again, I am predicating of this thought that it is true. So, by the identity claim, the thought expressed by “What John said at dinner is true” should be identical to the thought expressed by “Paris is in France.” But it seems wrong to say that “What John said at dinner is true” expresses the same thing that “Paris is in France” expresses. This appears to be a counter-example to the identity claim.
I do not want to come down one way or another on whether this is a genuine counter-example to the identity claim (although it seems to me that it is). Instead, I think we shouldn’t take Frege to be committed to that claim. To be sure, Frege does say “nothing is added to the thought by my ascribing to it the property of truth.” It is not my view that Frege does not explicitly commit himself to something like the claim that the thought P is identical to the thought T(P). What I want to suggest, however, is that Frege needn’t have said this, and that nothing he says elsewhere commits him to such a strong claim. Frege’s central point was not to make a positive claim about the identity of certain thoughts, but to make a negative claim about the sense of the truth-predicate. Namely, whether or not P and T(P) are the same thought, the latter “adds nothing” by way of assertoric force.
I have declared a spiritual war upon all coercion that restricts man’s creative activity. There are two kinds of coercion. One of them is physical[...].
The other kind of coercion is logical. We must accept self evident principles and the theorems resulting therefrom. This coercion is much stronger than the physical; there is no hope for liberation. No physical or intellectual force can overcome the principles of logic and mathematics. That coercion originated with the rise of Aristotelian logic and Euclidean geometry.
The concept was born of science as a system of principles and theorems connected by logical relationship. [...]
In the universe conceived in this way there is no place for a creative act resulting not from a law but from a spontaneous impulse [...].
The creative mind revolts against this concept of science, the universe and life. A brave individual, conscious of his value, does not want to be just a link in the chain of cause, but wants himself affect [sic] the course of events. This was always been [sic] the background of the opposition between science and art. [...]
He has two paths to choose from: either to submerge himself in scepticism and abandon research, or to come to grips with the concept of science based on Aristotelian logic. I have chose that second path. [...]
In striving to transform the concept of science based on Aristotelian logic I had to forge weapons stronger than that logic. It was symbolic logic that became such a weapon for me.
-J. Łukasiewicz, Farewell Lecture as Rector of Warsaw University, on March 7, 1918 (qtd. in R. Cignoli, 2007).
“A philosopher-mathematician loaded with explosives, lucid and reckless, resolute without optimism. If that’s not a hero, what is a hero?”
-Georges Canguilhem (speaking about Jean Cavaillès)
Rumor is, he was “highly impractical, eccentric, and very engrossed in his subject which was Philosophy”
In their 1976 paper, “Computer Science as Empirical Inquiry,” Allen Newell and Herbert A. Simon defined a “physical symbol system” in the following way:
A physical symbol system consists of a set of entities, called symbols, which are physical patterns that can occur as components of another type of entity called an expression (or symbol structure). Thus, a symbol structure is composed of a number of instances (or tokens) of symbols related in some physical way (such as one token being next to another). At any instant of time the system will contain a collection of these symbol structures. Besides these structures, the system also contains a collection of processes that operate on expressions to produce other expressions: processes of creation, modification, reproduction and destruction. A physical symbol system is a machine that produces through time an evolving collection of symbol structures. Such a system exists in a world of objects wider than just these symbol expressions themselves
They then proposed the following hypothesis: A physical symbol system has the necessary and sufficient means for general intelligent action
In tracing the historical roots of this hypothesis, they write:
The roots of the [Physical Symbol System] hypothesis go back to the program of Frege and of Whitehead and Russell for formalizing logic: capturing the basic conceptual notions of mathematics in logic and putting the notions of proof and deduction on a secure footing. This effort culminated in mathematical logic—our familiar propositional, first-order, and higher-order logics. It developed a characteristic view, often referred to as the “symbol game.” Logic, and by incorporation all of mathematics, was a game played with meaningless tokens according to certain purely syntactic rules. All meaning had been purged. One had a mechanical, though permissive (we would now say nondeterministic), system about which various things could be proved. Thus progress was first made by walking away from all that seemed relevant to meaning and human symbols. We could call this the stage of formal symbol manipulation.
But they have the history at least partly wrong. With respect to arithemtic, Frege was absolutely insistent that mathematics is not, as Newell and Simon write, “a game played with meaningless tokens according to certain purely syntactic rules.” The progress that Frege saw in his work on arithmetic was precisely that it didn’t walk away from “all that seemed relevant to meaning and human symbols.” Frege writes that
Many mathematicians are unclear about the import of formal arithmetic. Essentially, formal arithmetic seems to be regarded as meaningful arithmetic relieved of the obligation to supply a reference for the signs. In fact, the conception of numbers as figures is really used only at the outset, where that obligation is oppressive. Later one slides back unawares into meaningful arithmetic. And yet this formal conception has consequences which can be burdensome; so completely does it change arithmetic, from its very foundations up, that it hardly seems admissible to use the name `arithmetic’ for the formal as well as the meaningful study. Formal arithmetic can remain alive only by being untrue to itself. Its semblance of life is facilitated by the haste with which mathematicians usually hurry over the foundations of their science (if indeed they have any concern for them) in order to reach more important matters. Many things are omitted completely, others briefly touched on, nothing performed in detail.
“Does the mountain end here?” she asked, obnoxiously.
“What do you mean, here?” I responded, annoyed.
“Right here. Where I’m pointing?”
“What am I, an atlas?”
She pointed to the grasslands to the east of us. “Does the mountain extend over there?”
“Of course not!”
“Does it end over there?” pointing mid-mountain.
“Where’s my cheese steak. I’m sure I packed a cheese steak in the backpack. It’s gotta be somewhere.”
“Well, at what point does the mountain end?”
“Over there,” I said. I pointed with my whole hand, waving it a little bit.
Unfortunately, I didn’t find my cheese steak.
To the degree to which we regard our semantical methods as model-making (i.e., as a way of analyzing the notion of logical consequence for the object language) rather than as reality-describing (i.e., as analyzing the intended interpretation), fine-tuning the object language to bring it into conformity with our model may end up institutionalizing an artifact of the model that corresponds to no aspect of reality. I often think that my Platnoizing model-making is artificial, but I see nothing objectionable in being realistic about the artifacts qua artifacts. We model-makers love our artifacts. Models have their own reality, and the more we acknowledge that, the less likely we are to confuse the reality of the model with the reality it models. Model-making, by helping to articulate structure, can help to make it more acceptable that there is a reality behind questioned linguistic forms.
-David Kaplan, “Opacity,” in W.V. Quine (L. Hahn, ed.)
Scientific thinking, a thinking which looks on from above, and thinks of the object-in-general, must return to the “there is” which underlies it; to the site, the soil of the sensible and opened world such as it is in our life and for our body–not that possible body which we may legitimately think of as an information machine but that actual body I call mine, this sentinel standing quietly at the command of my words and acts.
-Maurice Merleau-Ponty, “Eye and Mind”
Merleau-Ponty says a number of interesting things in this quote. His main subject-matter is scientific thinking and how it characterizes human subjects. He isn’t saying that cognitive science, neuroscience, etc. are wrong. Instead, he’s saying that the models that they construct should at some point be reconnected with how we experience the world. The cognitive subject characterized in cognitive science should be brought into touch with the experiencing subject. There’s a lot of difficulty in unpacking what this means and whether it is right. Shouldn’t we expect that the “experiencing subject” is wrong? Is he just asking us to tie science down to folk psychology? I think it’s more complicated than any of this, although I’m not sure how to cash it out. In the end, I don’t think M-P thinks we ever manage to throw away the vestigial remnants of perceptual experience–or that this is really a problem. It’s not that we have some heuristic folk psychology that we just can’t give up. It’s much more all-encompassing. Even science involves “sensuous” perceptual experience.
The problem is when scientific models and formalisms are cut off from their standard/intended interpretations (like when we give non-standard models of arithmetic).* If we start to ignore the intended interpretation of models in cognitive science, then we will start to take the formalism to be more concrete than what it was meant to capture. Computational models, for example, are supposed to capture some very real, very salient feature of humans as they engage with the world. The human subject, embedded in an environment, is the intended model that cognitive scientific theories are trying to capture. No matter that those theories can be expressed mathematically. Arithmetic is supposed to deal with numbers dammit!
I am particularly interested in M-P’s talking about “that possible body which we may legitimately think of as an information machine.” A lot of people take M-P to suggest some sort of anti-computational picture. But he seems to think that we really can legitimately treat the body as an information machine (I don’t know if it’s historically accurate to translate this as “information-processing system,” and therefore as “computational system”). The point is that this is an abstraction or formalism. The intended model of the formalism is what M-P calls the “actual” body (presumably, how this should be understood is the subject of Phenomenology of Perception). Arithmetic is about numbers and cognitive science is about actual subjects!
*Ok, so M-P would never talk about standard/intended and non-standard/unintended models or interpretations, or about the semantics of formal systems or arithmetic, etc. etc. But I think this is all a perfectly plausible way to express his sentiment.
