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Workshop 5. Registration General. Constructive mathematics and models of type theory. Awodey CMU : Natural models of type theory.
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Kock Univ. Barcelona : Univalence in locally cartesian closed infinity-category. Palmgren Univ.
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Stockholm : Models of dependent types and dependently typed logic and their formalization in type theory. Voevodsky Princeton : B-systems - a link which connects the syntax of dependent type theories with their semantics. Awodey Natural models of type theory One thing missing from homotopy type theory is a notion of model that is both faithful to the precise formalism of type theory and yet general and flexible enough to be a practical tool.
Past attempts have relied either on highly structured categories corresponding closely to the syntax of type theory, such as the categories with families of Dybjer, or more natural and flexible categorical models based on homotopical algebra. The former are, however, somewhat impractical to work with semantically, while the latter must be equipped with unnatural and non-trivial coherence conditions. We present a new approach which combines these two strategies while retaining some of the advantages of each.
It is based on the observation that a category with families is the same thing as a representable natural transformation, in the sense of Grothendieck.
Varieties of constructive mathematics / Douglas Bridges, Fred Richman - Details - Trove
Ideas from Voevodsky and Lumsdaine-Warren are also used. In this talk I will discuss the issues which arise when one wants to construct a model category based on the traditional notion of paths as continuous maps from the interval, in a constructive framework such as Bishop constructive mathematics or extensional type theory. Such models would actually embody the intuitive idea that identity types are paths in a space, while preserving computability, and would also give us a plethora of examples via interpretations in models of constructive mathematics.
The talk will start from the 1-dimensional Seifert-van-Kampen Theorem for the fundamental group, then groupoid, and so to a use of certain strict multiple groupoids for higher versions. Indeed, the constructivist school has since been built on the foundation of what is known as intuitionistic logic , to distinguish it from traditional or "classical" logic. In the real world, if a teacher confronts you and says "Did you throw this paper aeroplane or didn't you? This is because either you threw it, or you didn't throw it, so a question which asks if you did either of these things must have a positive answer.
If A represents a statement such as "I threw the paper aeroplane", the only options seem to be A or "not A", the latter being the opposite of A in our example, "not A" is "I did not throw the paper aeroplane". This is known in classical logic as the law of the excluded middle , or LEM for short. What distinguishes intuitionistic logic from classical logic is the rejection of the LEM. This deceptively simple distinction disguises a profound difference in viewpoint about truth and meaning.
Someone who believes in classical logic is known as a Platonist or a realist. From the realist viewpoint, which is certainly the way you and I were taught mathematics and logic, the truth of a statement is determined by truth tables. The statement must be true because either I threw the aeroplane or I did not. But this argument rests on the idea that a statement is either true or false, which is known as bivalence. Intuitionists reject the validity of bivalence in mathematics — why? An intuitionist believes that a mathematical statement is made true or false by the construction of a proof.
In other words, a mathematical statement has no truth table value before a proof has been constructed. It is not so much that a statement exists in some weird middle ground between true and false, but rather that truth or falsity is a quality given to a statement by the construction of a proof by a human being using ideas and techniques of mathematics which are intuitively self-evident to the human mind.
As such, bivalence is rejected, and therefore so is the LEM. Brouwer used to provide what he claims to be a counterexample to the law of trichotomy; that law which states that every real number is either positive, negative or zero. Quite a claim, and here is how he made it. There will always be simple questions we can ask about , the answers to which we do not yet know. Fortunately, there are well-known algorithms for generating as many digits of as we like, so questions like "are there ever consecutive zeros in the decimal expansion of " may be answered in the affirmative one day, when we have sufficient computing power to follow the algorithms through to a large number of digits.
In this way we can always formulate a question such as this, be it zeros or sixes, or what have you. Let S be one of those statements, maybe "there are consecutive zeros in the decimal expansion of ". To most mathematicians and I suspect most of you, the statement "S or not S" is true. But the constructivist disagrees because of her rejection of the law of the excluded middle. The constructivist argues that it is a mistake to think of the decimal expansion of existing "out there" somewhere as an object independent of the human mind.
To the constructivist, the decimal expansion of only exists up to the truncated point to which it has been calculated by humans or our proxies, the computers. Suppose, he said, that we go on calculating until we find the first run of consecutive zeros. If it starts at the nth decimal place and n is odd, then let terminate at its nth digit. The key points to bear in mind are the following.
Constructivism (philosophy of mathematics)
We have a perfectly decent construction for : just calculate and let the two numbers be equal until we hit a run of zeros in , at which point follow the rules above. We do not know at present, and probably never will, whether such an n even exists. No matter, we can still discuss the quantity.
If no run of consecutive zeros ever appears then and. If such a run appears and begins at an even place in the expansion, then and ; if at an odd place then. The problem is that if after a billion years of calculating squadribblioblions of digits of we still have not found a run of consecutive zeros or whatever our simple statement says , then we still do not know whether is positive, negative, or zero, and are no closer whatsoever to an answer. But the law of trichotomy, from a classical standpoint, asserts that is either positive, negative, or zero, regardless of the fact that we will never know which.
In other words, there are situations in which we can never determine, even in principle, whether a number is positive, negative, or zero, and yet the law of trichotomy would insist that one must be true. What meaning can we give to this law, when such a simple construction reveals it to be effectively meaningless? A constructivist says that will be one of the three when if we ever get to a run of zeros, but is none of them beforehand.
The disturbing fact for realists is that this implies that mathematics is subjective and time-dependent, indeed that it is also dependent on such messy things as history and culture. This illustrates how firmly entrenched is the realist view that mathematical objects exist independently of the human mind, "out there" somewhere just waiting to be discovered, even in our every day conception of objects as fundamental as the real numbers.
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Intuitionism is radically antirealist: antirealist in that it claims mathematical objects only come into existence once they are constructed by a human mind a sad quirk of language that this is called "anti"realist , and radical since it seeks to recast all of mathematics in this light. We have looked here at a couple of telling examples of the problems constructivists see in the realist view of mathematics, and much more on this fascinating topic and its history can be found in the suggested reading below. But what has been the effect on modern mathematics of this revelatory and revolutionary viewpoint?
Surprisingly little, which is surprising in itself see above.
Related Varieties of Constructive Mathematics
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