M-theory and Mathematics If the framework of M-theory is the right path for a unification of gravity with quantum mechanics, is it obligatory to believe that, at a certain level, reality is not just easily describable by mathematics but actually transmogrifies into mathematics itself—i.e., we have learned of objects such as D-branes, Calabi–Yau manifolds, etc., but what are these objects except perfect (ideal) mathematical objects, not imperfect objects like shoes and plants?
 A: Many popular-science authors such as Stephen Hawking or Brian Greene would try to give you the impression that if M-theory passes all self-consistency checks, there will be one "Mathematically Inevitable Theory of Nature", M-theory. Not trying to diminish the immense proportion of the human achievement a "Theory of everything" would represent, the philosophical aspects are slightly more complicated than that. 
For instance, remember that gravity and quantum mechanics are not inevitable or "unimaginable to not exist". But their existence is the basic assumption in building modern Theories of everything! (In the case of M-theory, more like conjecturing, though.) We could very well have a universe with quantum principles but no gravity and a completely different self-consistent theory. Or we could have a world based on completely different logical principles. Or without quantum mechanics. So, what is the difference between our "gravity-quantum-theory" universe and the other universes that I mention? The difference is that the other universes are not realized while ours is.
A crash course in Aristotelianism
This idea runs deep in the European philosophical tradition (sorry for my ignorance of non-European philosophers), namely it is one of the building blocks of Aristotelianism. Essentially, Aristotelianism would state that there exist forms, such as geometrical or mathematical objects, but there is also substance which "fills the form". The evergreen motif of Aristotelianism is the conjecture of a "primal substance", materia prima, which is the only unique substance which "fills the form". The perceived difference between, say, earth and water is then only a question of the "form" (today we would say "structure") on an unperceived, microscopical level.
To state it differently, if there is only one type of substance, materia prima, then the fact that the "form", or geometrical structure, is "filled with the substance" plainly means that the form is realized rather than not. Being filled with materia prima or not is simply a question of existing or not.
Particles as forms
But let's turn back to physics. As is, the Standard model interpreted in Aristotelian terms would tell you that there are many realizable forms, these being all the particles of the Standard model. You may not think of the point-like electron or quark quite as a "form" because it has no geometrical extent, but the fact that you can attach numbers to all it's properties such as "charge", "mass" or "lepton number" would be enough for Aristotle or Thomas Aquinas to call it a form. Whether this "form" of the electron is realized or not is a question of substance. 
I think Aristotle or Thomas Aquinas would even delight in the modern formulation of particle physics. "Two electrons are indistinguishable," they would say, "so how come a second electron exists if there is no difference between them?" "Substance," Aquinas would lightly smile "substance was given to the very same form, the electron, to be realized twice".
However, there is a slight uneasiness to this, but this a soft uneasiness and, in my humble opinion, more caused by a conventional description and formalism rather than a true philosophical issue. You see, you could say that there is no materia prima binding all the particles because an electron is something different than a proton on a fundamental level. You use a different mathematical object to describe it, so the physical and philosophical object is also a fundamentally different one. Thus, for every particle there should be a different kind of substance. But one of the properties of "true substance" is that it cannot be destroyed. But every particle we know can be somehow transformed into a different one and thus cannot represent a substance of it's own. As is, the Standard model corresponds to a single materia prima.
Theory of everything
I am not going to milk it much longer, the common thread of all Theories of everything, along with their ambition to devour even cosmology and cosmic initial conditions, is


*

*To completely geometrize nature

*To make the whole of reality a single wholesome geometrical form


1) actually addresses the concerns in the last paragraph of the previous section. Yes, it is more elegant and beautiful to have it that way, but from a philosophical viewpoint it feels less relevant. You can see that 1) is somewhat fulfilled in string theory; particles are not points but configurations ("forms") of strings. 
Number 2) is essentially the quest to have every part of the theory as "inevitable". Initial conditions of the universe are "inevitable", physical laws are "inevitable" (btw. physical laws should also be considered to be part of the "reality-form"). And so on. If you sum it up, the result should be that under some assumptions, the "reality-form", "universe-form" or "world-form" is not modifiable, you cannot just change parts, at least not small ones.
But even if you do all this homework perfectly, and find these "inevitability" rules for the universe-form, you can ALWAYS change your assumptions. There will never be a unique universe-form or world-form or whatever we choose to call the Theory of everything. 
So what is the difference between the mathematical structure of the alternative universes, different world-forms, and our universe and world-form? Our world-form and the alternative world-form can be considered very much the same mathematical object. But can both be considered reality? Do they have the same status? I would say not. The difference between them is in the fact that one does get realized and one does not. Our world-form is a cup filled with primeval substance, materia prima, granted existence, whether the alternative world-forms are not.
So is reality mathematics? No. There is a line between reality and mathematics, the realized and unrealized. This line can be called "existence", "essence" or "substance".

A few side-notes:
Modern mathematics is actually just a string of words or symbols with allowed beginning called "axioms". This string of words or symbols is arranged according to some rules called "logic". Referencing previous theorems helps you not to have to state the string from the very beginning but reference to a known ending or branch of this string. In this way, you can effectively develop this "word game". 
But how do Dedekind cuts relate to a geometrical line? I find it really funny when some people would tell me that the mathematical object called $\mathbb{R}$ somehow is the geometrical line (I do understand the usage in common language, however.). But in fact, modern mathematics is truly vacuous, it does not refer to anything. 
An Aristotelian philosopher would tell you that the meaning of modern mathematics is in fact to purport "true forms" arising in these strings of words. Indeed, modern mathematics seems to be the most efficient tool in exploring true forms.  But these "true forms" such as a geometrical line have a meaning even without the modern axiomatic mathematical construction. In the previous paragraphs, I refer to mathematical and geometrical objects in this sense of "true forms" rather than "particular strings of symbols" (as is common with physicists and as was common with the whole scientific world before the twentieth century).
You should also know about the Platonic or Socratic tradition of European philosophy. In brief, they would find this whole endeavor to find a Theory of Everything petty.
