Oh, I'm willing to admit variously infinite numbers of applications of the rules... that's why transfinite induction doesn't bother me in the slightest.
But, my objection to the existence of abstract points is: what's the definition of a point? It's defined by what it does, by duck-typing. For instance, a point in R² is an ordered pair of reals. Now, you could say "an ordered pair (x,y) is the set {x,{x,y}}", but that's silly, that's not what an ordered pair is, it's just a construction that exhibits the required behaviour: namely, a constructor from two input values, and an equality axiom "(a,b)==(c,d) iff a==c and b==d". Yet, from a formal perspective at least, there are many models of those axiomata, and it's absurd to claim that any one of those is what a point "is" - far more sensible to say that the point "is" its axiomata. Since those axiomata essentially consist of a list of valid string-rewriting rules (like (a,b)==(c,d) |- a==c), they are directly and explicitly syntactic.
Perhaps, indeed, there is a system more fundamental to mathematics than syntactics - but given that the classes of formal languages even over finite strings are "variously infinite" (since language classes are equivalent to computability classes, by something Curry-Howardesque), it seems to me that, by accepting variously infinite strings and running-times, one should find that all mathematical systems are inherently syntactic in nature.
Sadly this is difficult to prove, as all our existing formal methods are themselves explicitly syntactic and thus anything we can express formally by current means, we can express as syntax. If materialistic and mechanistic ideas about the nature of consciousness are valid, then in fact any mathematics conceivable by human thought are susceptible to syntactic interpretation (for, ultimately, there exists a "validity" predicate over mathematical deductions, and assuming that validity predicate is constant in all factors other than the mathematical deduction itself (which assumption I believe to hold, as I am a Platonist), that predicate has a syntactic expression though possibly one derived via the physics of the brain). This does not, however, rule out the possibility that there are things we might want to call 'formal systems' which are not syntactic in nature. It is my belief - and nothing more than that - that such things do not exist.
These might be stupid questions, but I’m encouraged by a recent post to ask them:
But, my objection to the existence of abstract points is: what's the definition of a point? It's defined by what it does, by duck-typing.
Doesn’t that apply to syntactic methods, too? It was my understanding that the symbols, strings and transformation rules don’t quite have a definition except for duck typing, i.e. “symbols are things that can be recognized as identical or distinct from each other”. (In fact, in at least one of the courses I took the teacher explicitly sai...
Follow-up to: Syntacticism
I wrote:
In my experience, most people default1 to naïve physical realism: the belief that "matter and energy and stuff exist, and they follow the laws of physics". This view has two problems: how do you know stuff exists, and what makes it follow those laws?
To the first - one might point at a rock, and say "Look at that rock; see how it exists at me." But then we are relying on sensory experience; suppose the simulation hypothesis were true, then that sensory experience would be unchanged, but the rock wouldn't really exist, would it? Suppose instead that we are being simulated twice, on two different computers. Does the rock exist twice as much? Suppose that there are actually two copies of the Universe, physically existing. Is there any way this could in principle be distinguished from the case where only one copy exists? No; a manifest physical reality is observationally equivalent to N manifest physical realities, as well as to a single simulation or indeed N simulations. (This remains true if we set N=0.)
So a true description requires that the idea of instantiation should drop out of the model; we need to think in a way that treats all the above cases as identical, that justifiably puts them all in the same bucket. This we can do if we claim that that-which-exists is precisely the mathematical structure defining the physical laws and the index of our particular initial conditions (in a non-relativistic quantum universe that would be the Schrödinger equation and some particular wavefunction). Doing so then solves not only the first problem of naïve physical realism, but the second also, since trivially solutions to those laws must follow those laws.
But then why should we privilege our particular set of physical laws, when that too is just a source of indexical uncertainty? So we conclude that all possible mathematical structures have Platonic existence; there is no little XML tag attached to the mathematics of our own universe that states "this one exists, is physically manifest, is instantiated", and in this view of things such a tag is obviously superfluous; instantiation has dropped out of our model.
When an agent in universe-defined-by-structure-A simulates, or models, or thinks-about, universe-defined-by-structure-B, they do not 'cause universe B to come into existence'; there is no refcount attached to each structure, to tell the Grand Multiversal Garbage Collection Routine whether that structure is still needed. An agent in A simulating B is not a causal relation from A to B; instead it is a causal relation from B to A! B defines the fact-of-the-matter as to what the result of B's laws is, and the agent in A will (barring cosmic rays flipping bits) get the result defined by B.2
So we are left with a Platonically existing multiverse of mathematical structures and solutions thereto, which can contain conscious agents to whom there will be every appearance of a manifest instantiated physical reality, yet no such physical reality exists. In the terminology of Max Tegmark (The Mathematical Universe) this position is the acceptance of the MUH but the rejection of the ERH (although the Mathematical Universe is an external reality, it's not an external physical reality).
Reducing all of applied mathematics and theoretical physics to a syntactic formal system is left as an exercise for the reader.
1That is, when people who haven't thought about such things before do so for the first time, this is usually the first idea that suggests itself.
2I haven't yet worked out what happens if a closed loop forms, but I think we can pull the same trick that turns formalism into syntacticism; or possibly, consider the whole system as a single mathematical structure which may have several stable states (indexical uncertainty) or no stable states (which I think can be resolved by 'loop unfolding', a process similar to that which turns the complex plane into a Riemann surface - but now I'm getting beyond the size of digression that fits in a footnote; a mathematical theory of causal relations between structures needs at least its own post, and at most its own field, to be worked out properly).