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2 Additional argument for non-Boolean toposes

I am not sure I understand all remarks that Colin made, and I disagree with some of them, but I can comment on the idea of "multiverse". Let us consider the following positions:

1. There is a standard mathematical universe.
2. There are many mathematical universes.
3. There is a multiverse of mathematics.

These are supposed to be informal statements. You may interpret "universe" as "model of set theory ZF(C)" or "topos" if you like, but I don't want to fix a particular interpretation.

I would call the first position "naive absolutism". While many mathematicians subscribe to it on some level, logicians have known for a long time that this is not a very useful thing to do. For example, someone who seriously believes that there is just one mathematical universe would reject model theory as fiction, or at least brand it as a technical device without real mathematical content. Such opinions can actually slow down mathematical progress, as is historically witnessed by the obstruction of "Euclidean absolutism" in the development of geometry.

The second position is what you get if you take model theory seriously. Set theorists have produced many different kinds of models of set theory. Why should we pretend that one of them is the best one? You might be tempted to say that "the best mathematical universe is the one I am in" but this leads to an unbearably subjective position and strange questions such as "how do you know which one you are in?" At any rate, it is boring to stay in one universe all the time, so I don't understand why some people want to stick to having just one cozy universe.

I need to explain the difference between the second and the third position. By "multiverse" I mean an ambient of universes which form a structure, rather than just a bare collection of separate universes. The difference between studying a "collection of universes" and studying a "multiverse" is roughly the same as the difference between Euclid's geometry and Erlangen program -- both study points and lines but conceptual understanding is at different levels. Likewise, a meta-mathematician might prove interesting theorems about models of set theory, or he might consider the overall structure of set-theoretic models.

It should be obvious at this point that there cannot be just one notion of "multiverse". I can think of at least two:

• Set theory studies the multiverse of models of ZF. The structure is studied via notions of forcing, and probably other things I am not aware of.
• Topos theory studies the multiverse of toposes. The structure of the multiverse is expressed as a 2-category of toposes (and geometric morphisms).

I do not mean to belittle set theory, but in a certain sense topos theory is more advanced than set theory because it uses algebraic methods to study the multiverse (I consider category theory to be an extension of algebra). In this sense the formulation of forcing in terms of complete Boolean algebras by Scott and Solovay was a step in the right direction because it brought set theory closer to algebra. Set theorists should learn from topos theorists that transformations between set-theoretic models are far more interesting than the models themselves.

In the present context the question "classical or intuitionistic logic" becomes "what kind of multiverse". If multiverse is "an ambient of universes, each of which supports the development of mathematics" then taking our multiverse to be either too small or to big will cause trouble:

• if the multiverse is too small, we will be puzzled by its ad hoc properties and we will look in vain for overall structure (imagine doing analysis with only rational numbers),
• if the multiverse is too big, its overall structure will be poor and it will include universes whose internal mathematics is too far removed from our own mathematical experience (imagine doing analysis on arbitrary rings--I am sure it's possible but it's unlike classical analysis).

Topos theory gains little by restricting to Boolean toposes. I have never heard a topos-theorist say "I wish all toposes were Boolean". This is an argument Also, toposes occurring "in nature" (sheaves on a site) typically are not Boolean, which speaks in favor of intuitionistic mathematics.

An example of an ad hoc property in too small a multiverse occurs in set theory. We construct models of set theory by forming Boolean-valued sets which are then quotiented by an ultrafilter. What is the ultrafilter quotient for? The algebraic properties of Boolean-valued sets are hardly improved when we pass to the quotient, not to mention that it stands no chance of having an explicit description. A possible explanation is this: we are looking only at one part of the set-theoretic multiverse, namely the part encompassed by Tarskian model theory. Our limited view makes us think that the ultraquotient is a necessity, but the construction of Boolean-valued models exposes the ultraquotient as a combination of two standard operations (product followed by a quotient). We draw the natural conclusion: a model of classical first-order theory should be a structure that measures validity of sentences in a general complete Boolean algebra. The Boolean algebra $\lbrace 0,1 \rbrace$ must give up its primacy. What shall we gain? Presumably a more reasonable overall structure. At first sight I can tell that it will be easy to form products of models, and that these products will have the standard universal property (contrast this with ultraproducts which lack a reasonable universal property because they are a combination of a categorical limit and colimit). Of course, there must be much more.

1

I am not sure I understand all remarks that Colin made, and I disagree with some of them, but I can comment on the idea of "multiverse". Let us consider the following positions:

1. There is a standard mathematical universe.
2. There are many mathematical universes.
3. There is a multiverse of mathematics.

These are supposed to be informal statements. You may interpret "universe" as "model of set theory ZF(C)" or "topos" if you like, but I don't want to fix a particular interpretation.

I would call the first position "naive absolutism". While many mathematicians subscribe to it on some level, logicians have known for a long time that this is not a very useful thing to do. For example, someone who seriously believes that there is just one mathematical universe would reject model theory as fiction, or at least brand it as a technical device without real mathematical content. Such opinions can actually slow down mathematical progress, as is historically witnessed by the obstruction of "Euclidean absolutism" in the development of geometry.

The second position is what you get if you take model theory seriously. Set theorists have produced many different kinds of models of set theory. Why should we pretend that one of them is the best one? You might be tempted to say that "the best mathematical universe is the one I am in" but this leads to an unbearably subjective position and strange questions such as "how do you know which one you are in?" At any rate, it is boring to stay in one universe all the time, so I don't understand why some people want to stick to having just one cozy universe.

I need to explain the difference between the second and the third position. By "multiverse" I mean an ambient of universes which form a structure, rather than just a bare collection of separate universes. The difference between studying a "collection of universes" and studying a "multiverse" is roughly the same as the difference between Euclid's geometry and Erlangen program -- both study points and lines but conceptual understanding is at different levels. Likewise, a meta-mathematician might prove interesting theorems about models of set theory, or he might consider the overall structure of set-theoretic models.

It should be obvious at this point that there cannot be just one notion of "multiverse". I can think of at least two:

• Set theory studies the multiverse of models of ZF. The structure is studied via notions of forcing, and probably other things I am not aware of.
• Topos theory studies the multiverse of toposes. The structure of the multiverse is expressed as a 2-category of toposes (and geometric morphisms).

I do not mean to belittle set theory, but in a certain sense topos theory is more advanced than set theory because it uses algebraic methods to study the multiverse (I consider category theory to be an extension of algebra). In this sense the formulation of forcing in terms of complete Boolean algebras by Scott and Solovay was a step in the right direction because it brought set theory closer to algebra. Set theorists should learn from topos theorists that transformations between set-theoretic models are far more interesting than the models themselves.

In the present context the question "classical or intuitionistic logic" becomes "what kind of multiverse". If multiverse is "an ambient of universes, each of which supports the development of mathematics" then taking our multiverse to be either too small or to big will cause trouble:

• if the multiverse is too small, we will be puzzled by its ad hoc properties and we will look in vain for overall structure (imagine doing analysis with only rational numbers),
• if the multiverse is too big, its overall structure will be poor and it will include universes whose internal mathematics is too far removed from our own mathematical experience (imagine doing analysis on arbitrary rings--I am sure it's possible but it's unlike classical analysis).

Topos theory gains little by restricting to Boolean toposes. I have never heard a topos-theorist say "I wish all toposes were Boolean". This is an argument in favor of intuitionistic mathematics.

An example of an ad hoc property in too small a multiverse occurs in set theory. We construct models of set theory by forming Boolean-valued sets which are then quotiented by an ultrafilter. What is the ultrafilter quotient for? The algebraic properties of Boolean-valued sets are hardly improved when we pass to the quotient, not to mention that it stands no chance of having an explicit description. A possible explanation is this: we are looking only at one part of the set-theoretic multiverse, namely the part encompassed by Tarskian model theory. Our limited view makes us think that the ultraquotient is a necessity, but the construction of Boolean-valued models exposes the ultraquotient as a combination of two standard operations (product followed by a quotient). We draw the natural conclusion: a model of classical first-order theory should be a structure that measures validity of sentences in a general complete Boolean algebra. The Boolean algebra $\lbrace 0,1 \rbrace$ must give up its primacy. What shall we gain? Presumably a more reasonable overall structure. At first sight I can tell that it will be easy to form products of models, and that these products will have the standard universal property (contrast this with ultraproducts which lack a reasonable universal property because they are a combination of a categorical limit and colimit). Of course, there must be much more.