The materialists pull everything down from the sky and out of the invisible world onto the earth as if they wanted to clench rocks and oak trees in their fists. They grasp them, and stubbornly maintain that the only objects that exist are those that are tangible and comprehensible. They believe that the physical existence of an object is existence itself, and look down smugly on other people — those who acknowledge another area of existence separate from the physical. But they are totally unwilling to listen to another point of view. (Plato, Sophisles)

Philosophical and mathematical correspondence

The Theory of Meaning

In Unit Operations, Ian Bogost argues that similar principles underlie both literary theory and computation, proposing a literary-technical theory that can be used to analyze particular videogames. Moreover, this approach can be applied beyond videogames: Bogost suggests that any medium -- from videogames to poetry, literature, cinema, or art -- can be read as a configurative system of discrete, interlocking units of meaning, and he illustrates this method of analysis with examples from all these fields. The marriage of literary theory and information technology, he argues, will help humanists take technology more seriously and hep technologists better understand software and videogames as cultural artifacts. This approach is especially useful for the comparative analysis of digital and nondigital artifacts and allows scholars from other fields who are interested in studying videogames to avoid the esoteric isolation of "game studies." The richness of Bogost's comparative approach can be seen in his discussions of works by such philosophers and theorists as Plato, Badiou, Zizek, and McLuhan, and in his analysis of numerous videogames including Pong, Half-Life, and Star Wars Galaxies. Bogost draws on object technology and complex adaptive systems theory for his method of unit analysis, underscoring the configurative aspects of a wide variety of human processes. His extended analysis of freedom in large virtual spaces examines Grand Theft Auto 3, The Legend of Zelda, Flaubert's Madame Bovary, and Joyce's Ulysses. In Unit Operations, Bogost not only offers a new methodology for videogame criticism but argues for the possibility of real collaboration between the humanities and information technology.

Unit Operations: An Approach to Videogame Criticism

In this shmt passage, van Gogh provides us with a clear example of how complex it is to see. TO see an object or a certain state of affairs properly, it is not merely enough to stand in front of it and look ln the quoted passage, the same tones and colors that van Gogh used to see suddenly appeared to him in a new light We do not know exactly how Mauve taught him to see what he was not able to see before Perhaps, both of them were standing in front of a painting and Mauve, holding his btush, pointed to some colours while explaining their contrast and tone differences. Pointing and words may have made visible, for the first time to Vincent van Gogh, something new something that had escaped him until then Mauve's lesson is an example of a process of objectification: that is, etymologically speaking, a process aimed at bringing something in front of someone's attention or view Mauve's signs (e g the words and gestures that he might have used to make his point about coloms and tones) are part of what I have termed elsewhere (Radford, 2003) semiotic means of objectification e g objects, artifacts, linguistic devices and signs that are intentionally used by individuals in social processes of meaning production, in order to achieve a stable fmm of awareness, to make apparent their intentions and to carry out their actions In learning mathematics, our contempotaty students face a situation that is not so very different from van Gogh's. Students also need to learn to see beyond etude perception and to find something that was unnoticed previously e g the center of gravity of a triangle, an irrational nmnber such as >/2 or the general term of a pattern. But to make the mathematical objects somehow apparent even the most accmate ostensive gesture would not be good enough, for, as Gottlob Frege observed, mathematical objects are neither palpable nor directly perceivable How then are we going to deal with objects that cannot be directly perceived, such as nmnbers and other mathematical objects? Frege answered: with symbols (1892/1977, p. 144) But Frege's insightful answer leads inunediately to a trickier, more difficult question ln what sense do mathematical objects become known through signs? Can we equate the conceptual objects with the symbols representing them? Frege categorically answered, no, contending that we have to avoid the danger of confusing the objects with their symbols

methods

The Seen, the Spoken, and the Written: A Semiotic Approach to the Problem of Objectification of Mathematical Knowledge.

1. The Problem Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. Never mind whether Frege was right about this. I want to address a different question: Does Frege’s position on arithmetic really contradict Kant’s? I do not deny that Frege endorsed (F) Arithmetic is reducible to logic

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Frege, Kant, and the Logic in Logicism

ion. What would be left if we abstracted from all contentful functionand objectsenses in a Fregean thought would be nothing but a pattern of functional applications and abstractions. For example, instead ofions. For example, instead of for every two integers, there is a third that is greater than their sum, i.e., (∀x)(∀y)((Ix & Iy)⊃(∃z)(Iz & z>x+y)), which we might rewrite in functional notation as ∀(λx∀(λy⊃(&(I(x),I(y)),∃(λz>(z,+(x,y)))))), we would have the “form” Φ(λxΦ(λyζ(θ(π(x),π(y)),Ψ(λzκ(z,μ(x,y)))))). But at this level of abstraction the logical structure of the claim is no longer fully in view, since we’ve “abstracted from” the logical constants. We could continue to maintain that logic is unrestrictedly formal, but only at the cost of confining logic to the meager resources of function application and abstraction, and removing conjunction, negation, quantification, and other notions generally regarded as logical from its purview. It seems more reasonable to follow Frege in denying that logic is unrestrictedly formal.25 For Frege, then, logical notions cannot be distinguished from non-logical ones on by generic features of their semantic role. Both the logical notion identity and the non-logical notion is taller than are first level functions with two arguments. They differ only in their graphs: One puzzle: in the long passage quoted from FG2, Frege lists subsumption (the falling of an object under a concept or a first level concept under a second level concept, PW:193, 213) as one of the notions proper to logic. But subsumption is not a concept or relation, but the notion of functional application itself (or rather a restriction of that notion, since not all functions are concepts). What is more, subsumption does not correspond to any symbol in the Begriffsschrift, so logical relations that depend on it would be preserved under arbitrary category-preserving translations. It is therefore hard to see how it belongs with identity, negation, and concept subordination in Frege’s list. CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 157 X Y X is taller than Y X is identical with Y George Bush Jesse Jackson False False Odysseus Penelope True False George Bush George Bush False True etc. etc. etc. etc. Similarly, both the universal quantifier and the second level concept some philosophers are second level functions with one argument. The line between logical and non-logical notions must therefore appeal to specific features of items within each semantic category (perhaps permutation invariance or something similar). For Kant, by contrast, there is a generic difference in semantic functioning between logical and non-logical vocabulary. Notice that Frege’s argument against the 3-formality of logic in FG2 does not depend on his commitment to logical objects and his logicist program (though surely it is motivated, at least in part, by these commitments). One could consistently hold that logic has its own contentful concepts (negation, identity, concept subordination, etc.) while denying that objects can be given through logical means alone. This observation will be important in section 5.4, when we ask where we should stand in the debate between Frege and Kant on the nature of logic, given that most philosophers now reject the idea that logic has its own objects. 5.2.6 “The most general laws of truth” To hold that logic is 1-formal but not 3-formal, as Frege does, is to reject Kant’s Thesis. On Frege’s view, the laws of logic are both constitutive norms for thought as such and informative about very general features of the “nature of things.” Frege devotes considerable attention (from 1893 on) to explaining how this can be so. Frege’s main concern is to defuse an argument that would establish Kant’s Thesis on quite general grounds (grounds independent of any specifically Kantian doctrines). The argument begins with the observation that “law” is used in two senses, which we might call “normative” and “descriptive:” “In one sense a law asserts what is; in the other it CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 158 prescribes what ought to be” (GGZ:xv). A normative law prescribes what one ought to do or provides a standard for the evaluation of one’s conduct as good or bad. Public statutes, moral codes, and aesthetic canons are all normative laws. A descriptive law, on the other hand, describes counterfactually robust regularities in the order of things. Laws of physics, biology, and geometry are descriptive laws. Whether a law is normative or descriptive is an intrinsic feature of its content: no law can be both. The argument Frege is concerned to rebut simply applies this exclusive categorization to the laws of logic. If these laws are normative for thought as such (1-formal), then they cannot be descriptive: they cannot say anything about how things are. Hence, if they are 1-formal, they must also be 3-formal. Against this line of thought, Frege wants to have it both ways: he wants to conceive logical laws both as prescribing how one ought to think and as saying how things are. He does not think that logical laws are explicitly prescriptive as to their content (Ricketts 1996:127). They have the form “such and such is the case,” not “one should think in such and such a way”: The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions, which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature are general features of what happens in nature, and occurrences in nature are always in accordance with them. It is rather in this sense that I speak of laws of truth [i.e., laws of logic]. Here of course it is not a matter of what happens but of what is. (Th:58) But although logical laws are not prescriptive in their content, they imply prescriptions and are thus prescriptive in a broader sense: “From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring” (Th:58, emphasis added). Hence Any law asserting what is, can be conceived as prescribing that one ought to think in conformity with it, and is thus in that sense a law of thought. This holds for laws of geometry and physics no less than for laws of logic. The latter have a special title to the name “laws of thought” only if we mean to assert that they are the most general laws, which prescribe universally the way in which one ought to think if one is to think at all. (GGZ:xv) Frege’s line of thought here is subtle enough to deserve a little unpacking. Consider the statement “the white King is at C3.” Though the statement is descriptive in its content, it CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 159 has prescriptive consequences in the context of a game of chess: for instance, it implies that white is prohibited from moving a bishop from C4 to D5 if there is a black rook at C5. Now instead of chess, consider the “game” of thinking about the physical world. As in chess, “moves” in this game—thoughts—can be assessed as correct or incorrect. Thoughts about the physical world are correct to the extent that they accurately depict the way the world is. Laws of physics (for instance, Maxwell’s equations) are descriptive laws; they tell us how the physical world is. But in the context of the “game” or activity of thinking about the physical world, they have prescriptive consequences: one ought not make judgments that conflict with them. In so far as one’s activity is to count as thinking about the physical world, it must be assessable as correct or incorrect (true or false) by reference to the laws of physics.26 In this sense, the laws of physics provide constitutive norms for the activity of thinking about the physical world. This is not to say that one cannot think wrongly about the physical world: one’s thoughts need not conform to the norms provided by the laws of physics; they need only be assessable in light of them. Nor is it to say that one must be aware of these laws in order to think about the physical world. The point is simply that to count someone as thinking about the physical world is ipso facto to hold her thoughts assessable for truth or falsity by their agreement or disagreement with the laws of physics. Someone whose thoughts were not so assessable could still be counted as thinking, but not as thinking about the physical world. It is in this sense that Frege holds that a law of physics “. . . can be conceived as prescribing that one ought to think in conformity with it, and is thus in that sense a law of thought.” On Frege’s view, then, laws of physics cannot be distinguished from laws of logic on the grounds that the former are descriptive and the latter prescriptive. Both kinds of laws are descriptive and have prescriptive consequences. They differ only in the activities for which they provide constitutive norms. While physical laws provide norms for thought about the physical world, logical laws provide norms for thought as such. To count an activity as There are of course other dimensions of correctness of thought besides truth and falsity: e.g., standards for proper justification. CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 160 thought about the physical world is to hold it assessable in light of the laws of physics; to count an activity as thought at all is to hold it assessable in light of the laws of logic. (As in the case of the laws of physics, there is no implication that thought must conform to these laws in order to count as thought: the point is that they must be assessable in light of these laws.)27 Frege often casts the difference between logical laws and laws of the special sciences as a difference in generality : logical laws are more general in the sense that they “. . . prescribe universally the way in which one ought to think if one is to think at all” (GGZ:xv), as opposed to the way in which one ought to think in some particular domain (cf. PW:1456). Thus, in his 1897 “Logic” manuscript, Frege writes: Like ethics, logic c

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Philosophical and mathematical correspondence

Citations

The Theory of Meaning

Unit Operations: An Approach to Videogame Criticism

The Seen, the Spoken, and the Written: A Semiotic Approach to the Problem of Objectification of Mathematical Knowledge.

Frege, Kant, and the Logic in Logicism

What does it mean to say that logic is formal

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