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I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, relativity, etc.

But after years of post-graduate study in mathematics, I've become a bit disillusioned with the way mathematics is done, and I often feel that much of it has become merely a logical game with objects that have no meaning outside of the confines of the game; e.g., existence of enough injectives/projectives, existence of bases for any vector space, etc.

I am not really an applied person---I love abstract theories such as category theory---but I would like to feel that the abstractions have some meaning outside of the logical game. To me it seems that using non-constructive proof techniques divorces the theory from reality.

Essentially, my question boils down to the following:

1. If I adopt a constructive foundation (like say an intuitionistic type theory), is there any (apparently) insurmountable difficulty in modelling the physical universe and the abstract processes that emerge from it (quantum mechanics, relativity, chess, economies, etc.).




2. Is there any physical/computational justification for assuming excluded middle or some sort of choice axiom?

It depends on what you want out of your science!

If you want your relativity theory to include dramatic singularity theorems — then probably you will want excluded middle to prove them.

If you want your economics to use fixed-point theorems with short and conceptual proofs — then probably you will want excluded middle in those proofs.

But if what you want is predictions and calculations within specified error limits, then you won’t need excluded middle to do them: the arithmetic of rationals is decidable.

Similarly in chess, most of the instances of excluded middle that would occur to you will be constructively valid from the finitary nature of the situation.

So you can use examples from mathematical physics or mathematical economics in discussing constructive math. However, the existing modeling practices are diverse enough that they don’t provide a clear and non-circular justification for adopting or abandoning the logical principles at issue.

In ZF, you can explicitly define a well-ordering of the entire constructible universe — so the axiom of choice is the theorem of choice if you reject the existence of noncosntructible sets (i.e. adopt the axiom of constructibility).

You can even get the axiom of choice as a theorem or even a triviality in somewhat more constructive settings. For example, given a family of types $X_alpha$, the translation of the proposition $forall alpha exists x : x in X_alpha$ into dependent type theory via the propositions-as-types paradigm is precisely the type of choice functions for the family, and so the assertion that this proposition is true is literally the same thing as specifying a choice function.

In a universe without the excluded middle, one distinguishes between the set $Omega$ of truth values and the set ${ bot, top }$ consisting of false and true. While predicates are generally $Omega$-valued, one still likes to find ${ bot, top }$-valued predicates where one can, since they are nicer to work with.

One can appreciate this sort of idea even in a purely classical context. For example, consider the category of topological spaces.

If $S = { bot, top } $ is a two-point space, define an "S-valued predicate" on a space $X$ to mean a continuous function $f : X to S$. Any such function has a corresponding subspace $f^{-1}({top}) subseteq X$.

If $S$ has the indiscrete topology, then $S$-valued predicate are the same thing as functions $X to { bot, top }$, so these are just boolean predicates on the points of $X$.

If $S$ is the Sierpinski space and $top$ is the open point, then $S$-valued predicates correspond to open subspaces of $X$, and thus describe some sort of "open property" that respects the topology of the space. Alternatively, taking $top$ to be the closed point gives a notion of a "closed" property.

If $S$ has the discrete topology, then $S$-valued predicates on a space $X$ correspond to a property on the connected components of $X$. While these predicates are less common, they tell you something very strong about the space.

My general philosophy on these things is very heavily influenced by the idea of an internal language — in particular the internal language of a topos, or more general categories.

So, while you might want to do some work in some weaker setting, the most expedient approach to the foundations is:

• Formalize of abstract mathematics in the way that's easiest to do work in


• Using this convenient ambient mathematical setting, define/construct the types of universes you want to work in, and develop their theory


• Flesh out the internal language of this universe

As an example, you mention an interest in "intuitionistic type theory" — it turns out, for example, that (intuitionistic) dependent type theories = locally cartesian closed categories. If you also insist on a type of truth values, you get precisely the notion of "elementary topos".

For the purposes of doing classical physics where you've defined a (topological) state space, or at least hypothesized one without actually constructing one, then a particularly appealing construction is the topos of sheaves on state space.

The internal language of this topos allows you precisely express various ideas I've seen thrown about based on 'physical intuition'.

For example, the idea that "physical variables don't have exact values" can be expressed by observing you can have an internal real number (i.e. some real-valued physical observable) $X$ with the property that $X = r$ is identically false for every ordinary real number $r$. In fact, this is a typical property for internal real numbers to have; $X = r$ will always be identically false unless there is some open subset $U$ of state space on which $U$ is constant.

Your question appeals to me in its clarity and relevance. I would love to give a similar clear answer, but as we stand in math and physics I believe that matters are not so clear-cut.

As pointed out in the comments above, no mathematics suffices to describe reality. But in physics we try nonetheless to describe parts of reality, and especially patterns that we believe to observe. For these patterns we also seek explanations in such a way that it helps us to delve deeper into the physical world. By delving deeper I mean that we gain more traction over the things we observe. For instance we now have 'access' to particle-physics processes, which enables us to study and observe yet more deeply the mysteries of the Planck-scale world.

Mathematics can also be seen as the science of patterns and (cor)relations. For physics, I believe it doesn't intrinsically matter which part of mathematics we use to describe a pattern, as long as the mathematical description fits our observations and helps us to gain more traction [I would like to say: 'helps us to gain more understanding', but history shows that what we in our time believe to be understanding will probably be labeled misunderstanding by generations to come...].

Now constructive mathematics offers a very beautiful and fruitful 'new' framework/perspective to look at the physical world. I personally believe that, in one very fundamental way, intuitionistic mathematics is more suited for physics than classical mathematics BUT...

in many other ways, classical mathematics offers the mind 'easy' approaches to explore new patterns and (cor)relations, precisely through not having to worry about feasibility, reality etc. This 'freedom to ignore boundaries' is very important I believe in ANY science, and I often find it distressing to see that unorthodox creativity and real originality have a hard time in our academic communities.

So, summarizing I would say that

a) there are no insurmountable obstacles to phrase our current physics in intuitionistic mathematics, but it would need hard work to do so

b) intuitionistic mathematics has a fundamental perspective to offer, both unifiying and simplifying in nature, which I believe would go a long way in helping us delve deeper, so this work would be justified

c) it would be unwise to abandon classical mathematics for its ('unjustified') dreamlike abstractions, on the contrary we should embrace any mental dreams as long as they offer simplicity, beauty, mystery, etc. The drawback that I personally perceive with classical math in relation to physics is that the tacit acceptance of LEM and impredicativity often lead to mathematical machinery that is more complicated than necessary. And as I think to learn from history, simplicity and elegance are the gateway to progress in physics.

To answer your second subquestion: LEM plays an important role in so-called 'discrete' math systems in constructive mathematics. 'Discrete' then signifies that the equality of objects is decidable. For instance the algebraic numbers are a discrete subset of the complex numbers, and this gives a lot of constructive traction to 'constructivize' all sorts of classical theorems about the complex numbers. But in general of course LEM fails hopelessly. Certain choice principles are somewhat generally accepted in constructive mathematics, on the other hand most choice principles can be avoided by choosing careful definitions which incorporate the extra information necessary.

Regarding your first question,

.. modelling the physical universe and the abstract processes that emerge from it ..

is in my understanding, a science which has a name: it is called theoretical physics (some may say mathematical physics) and it is as wide as implied by the definition ;) Imo, it does not really depend on whether you will adopt a constructive mathematical approach or not.

Of course, since i feel it is phenomenology that we are speaking about here, "constructiveness" is apparently important for a physical theory, if it is to be tested against the experiment after all. But i do not see how "adopting a constructive foundation" itself would guarantee anything at all.

Maybe it would be relevant at that point, to recall the axiomatic foundation of the special relativity, as outlined in the celebrated article "On the electrodynamics of moving bodies". It is a theory which yields solid experimental evidence. Far from being any kind of expert in logic or methodology, i understand the argumentation of the article as a sequence of implications derived from the two initial postulates of special relativity. Maybe, there could have been different choices of necessary conditions. Possibly leading to comparable but different "theories". Formally or conceptually; this may be a matter of research on the foundations. Is this constructive in some sense? deriving some set of necessary conditions, stemming from some postulates, and building a model around this? Whose validity will be finally decided neither upon its foundational assumptions or its logical coherence but rather against experimental measurements. (Well it is an apparently successful theory of physics that you get like this but are its mathematical foundations regarded constructive?)

Regarding your second question, on whether there is any

physical/computational justification for assuming excluded middle or some sort of choice axiom

i do not know on the physical/computational assumptions on the excluded middle, but the use of infinite dimensional vector spaces and their bases is an everyday tool even in elementary quantum mechanics -let alone the various field theories facing the problem of second quantization.