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Topoi: The Categorial Analysis of Logic
A classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally.
Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry.
Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry.
592 pages, Paperback
First published November 1, 1979
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Displaying 1 - 6 of 6 reviews
April 27, 2013
Update: 2013, and this is the text I keep coming back to. Do I understand it any better than in 2009? Possibly the opposite :-/
(Page updates will refer to the reread I'm currently doing in connection with some Badiou - Lacan - Merleau-Ponty work.)
Conjecture: We can think of a category as a means of studying relations without a fixed medium, the logical equivalent of an aetherless physics.
[RG1:] One of the primary perspectives offered by category theory is that the concept of arrow, abstracted from that f function or mapping, may be used instead of the set-membership relation as the basic building block for developing mathematical constructions, and expressing properties of mathematical entities. Instead of defining properties of a collection by reference to its members, i.e. internal structure, one can proceed by reference to its external relationships with other collections.
[RG2:] A category may be thought of in the first instance as a universe for a particular kind of mathematical discourse. Such a universe is determined by specifying a certain kind of "object" and a certain kind of "arrow" that links different objects.
[RG3:] [T:]he individuals of a topos may be thought of as 'generalised' sets and functions that may well be nonextensional.
Remark: The fundamental tradeoff seems to be between a capacity for intensional discrimination and a too-positively defined closure. Reflective discrimination is bought at the price of scope; the price of intension is extent. . Exactly the same as what happens in the Penrose setting, and with nonclassical logics (relevance). We can take this in terms of dual negation.
[RG4:] Within any mathematical theory, isomorphic objects are indistinguishable in terms of that theory. The aim of that theory is to identify and study constructions and properties that are "invariant" under the isomorphisms of the theory... Category theory then is the subject that provides an abstract formulation of the idea of mathematical isomorphism and studies notions that are invariant under all forms of isomorphism. In category theory, "is isomorphic to" is virtually synonymous with "is".
[RG5:] In Grp and Mon... the initial objects are the same as the terminal ones (and so the equation 0 = 1 is "true up to isomorphism"). An object that is both initial and terminal is called a zero object. Set has no zero's. The fact that Grp and Mon have zeros precludes them, as we shall see, from being topoi.
Remark: I may want to take the zero object as an index of ideality. Existence, on the other hand (pure extensionality) is what opens[?:] a gap of presence between the initial and the terminal.
[RG6:]
Remark: Injection is indistinguishable from inclusion, up to isomorphism. What is it that lets us speak of existence as anything other than equality up to isomorphism? Two answers. 1) Pure multiplicity: Socrates and Meno are two, no matter how isomorphic they are with respect to the form of rationality. But what if spatiotemporality itself (the idea of khora) is taken up as one of the terms we place in logical relation? 2) An intention, an intension. (We're nearing the point of productive ambiguity between these.) To say that A is merely isomorphic and not identical to B can be to say that we have an intended interpretation of A, a role that we assign to it, a place, that differs from the intended interpretation / role / place of B. (Note the return of place, khora, in both cases). But in that case, as DH elucidates helpfully about the G-sentence, we can look at the matter in two ways again. Isomorphism can be what fails to distinguish intensions (in that sense, belonging to the gesture of transcendental philosophy, which seeks the meaning of the phenomenon in the intentional act), but ismorphism can also be a means of getting out of the straightjacket of transcendental philosophy: the discovery of an unexpected isomorphism brings us closer to the real by a path that runs diagonally across intended interpretations. In both cases, perhaps we can say that the notion of isomorphism, as part of the conceptual pair identity/isomorphism, helps to think the difference and relation of intension and extension...
Identity as a power of identification vs. identity as the limit of a power of differentiation, a point of impotence of some discriminant.
The diagram on 89 should look familiar to those who follow AB! "This construction looks rather special, but it is to be found whenever there are functions." A brilliant reversal! We use the ambiguity, the loss of information in the original function, which need not be one-to-one, to discover a partition of disjoint classes in the original domain, as if we learned something of untouched being through our ignorance of it! This is why AB uses this pivot of the indexing relation, like the divided line, as the engine of a phenomenological ontology. Compare "transcendental affinity" = what we learn about the noumenon simply because of the way it was bundled into the phenomenon.
(Page updates will refer to the reread I'm currently doing in connection with some Badiou - Lacan - Merleau-Ponty work.)
Conjecture: We can think of a category as a means of studying relations without a fixed medium, the logical equivalent of an aetherless physics.
[RG1:] One of the primary perspectives offered by category theory is that the concept of arrow, abstracted from that f function or mapping, may be used instead of the set-membership relation as the basic building block for developing mathematical constructions, and expressing properties of mathematical entities. Instead of defining properties of a collection by reference to its members, i.e. internal structure, one can proceed by reference to its external relationships with other collections.
[RG2:] A category may be thought of in the first instance as a universe for a particular kind of mathematical discourse. Such a universe is determined by specifying a certain kind of "object" and a certain kind of "arrow" that links different objects.
[RG3:] [T:]he individuals of a topos may be thought of as 'generalised' sets and functions that may well be nonextensional.
Remark: The fundamental tradeoff seems to be between a capacity for intensional discrimination and a too-positively defined closure. Reflective discrimination is bought at the price of scope; the price of intension is extent. . Exactly the same as what happens in the Penrose setting, and with nonclassical logics (relevance). We can take this in terms of dual negation.
[RG4:] Within any mathematical theory, isomorphic objects are indistinguishable in terms of that theory. The aim of that theory is to identify and study constructions and properties that are "invariant" under the isomorphisms of the theory... Category theory then is the subject that provides an abstract formulation of the idea of mathematical isomorphism and studies notions that are invariant under all forms of isomorphism. In category theory, "is isomorphic to" is virtually synonymous with "is".
[RG5:] In Grp and Mon... the initial objects are the same as the terminal ones (and so the equation 0 = 1 is "true up to isomorphism"). An object that is both initial and terminal is called a zero object. Set has no zero's. The fact that Grp and Mon have zeros precludes them, as we shall see, from being topoi.
Remark: I may want to take the zero object as an index of ideality. Existence, on the other hand (pure extensionality) is what opens[?:] a gap of presence between the initial and the terminal.
[RG6:]
Remark: Injection is indistinguishable from inclusion, up to isomorphism. What is it that lets us speak of existence as anything other than equality up to isomorphism? Two answers. 1) Pure multiplicity: Socrates and Meno are two, no matter how isomorphic they are with respect to the form of rationality. But what if spatiotemporality itself (the idea of khora) is taken up as one of the terms we place in logical relation? 2) An intention, an intension. (We're nearing the point of productive ambiguity between these.) To say that A is merely isomorphic and not identical to B can be to say that we have an intended interpretation of A, a role that we assign to it, a place, that differs from the intended interpretation / role / place of B. (Note the return of place, khora, in both cases). But in that case, as DH elucidates helpfully about the G-sentence, we can look at the matter in two ways again. Isomorphism can be what fails to distinguish intensions (in that sense, belonging to the gesture of transcendental philosophy, which seeks the meaning of the phenomenon in the intentional act), but ismorphism can also be a means of getting out of the straightjacket of transcendental philosophy: the discovery of an unexpected isomorphism brings us closer to the real by a path that runs diagonally across intended interpretations. In both cases, perhaps we can say that the notion of isomorphism, as part of the conceptual pair identity/isomorphism, helps to think the difference and relation of intension and extension...
Identity as a power of identification vs. identity as the limit of a power of differentiation, a point of impotence of some discriminant.
The diagram on 89 should look familiar to those who follow AB! "This construction looks rather special, but it is to be found whenever there are functions." A brilliant reversal! We use the ambiguity, the loss of information in the original function, which need not be one-to-one, to discover a partition of disjoint classes in the original domain, as if we learned something of untouched being through our ignorance of it! This is why AB uses this pivot of the indexing relation, like the divided line, as the engine of a phenomenological ontology. Compare "transcendental affinity" = what we learn about the noumenon simply because of the way it was bundled into the phenomenon.
June 19, 2024
They say you learn category theory with your second book, this one was mine. I recommend you try your hand with a more frustrating book (like Emily Riehl's Category Theory in Context), and then get this one so you can appreciate how lucid it is. I gave it 5 stars for the material and conversational tone. I wish more mathematics textbooks were written in this particular style, the current Bourbaki structure of "theorem, proof, theorem, proof" works fine for textbooks with an instructor in mind, but not for those of us who wish to self-study. I think that for that reason this is one of the most popular mathematics textbooks, I see it at every bookstore with a mathematics section (those are ever shrinking by the way!)
December 7, 2007
Well, I wanted to get category theory straight in my head, and with this accomplished that goal...but at what price? A fairly turgid work, but perhaps that's necessary for handling this field.
December 15, 2015
What Goldblatt lacks in elegance and concision he mostly makes up for in scope.
January 20, 2023
对用集合论作为数学的逻辑基础的标准做法进行了精彩的评论
December 8, 2023
Oh, what a journey!
Gotta learn to do sketches of elephants next.
Gotta learn to do sketches of elephants next.
Displaying 1 - 6 of 6 reviews