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Mathematical formalism regards mathematics as a syntactic matter, where symbols are manipulated according to rules and the symbols need not have any meaning. I am wondering though whether it has anything to say about if one then actually does assign meaning to a formal system.

So for example, Wikipedia notes that "Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).[own emphasis]". Does formalism itself have any opinion on how the interpretation can be given? Does it not care? Or do formalists perhaps even think that we shouldn't try to give any interpretations to formal systems? Put differently, can I understand semantics from a formalist perspective, or do I have to consider different schools of thought if I want to talk about semantics. Thanks.

  • Think that question was a bit different. But thank you for the references. I haven't heard of those things, will google asap. – Neil Aug 6 at 14:24
  • Colorless green ideas sleep furiously. – What does it mean for an abstract concept to have an opinion on anything? – xiota Aug 7 at 23:14
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I'm going to kind of ad lib here, but as a formalist, hopefully at least something of interest comes out of this.

For a formalist, Semantics is how mathematics gets its practical application. When I say something like "one plus one equals two", I am discussing something logical and syntactic, but when I say "one orange plus one orange equals two oranges", I'm discussing oranges. If I have some theorem about functions over the domain of a three dimensional vector space, I'm talking logic and syntax; if I say that this theorem allows me to show a feature of how I can manipulate the orange in physical space, the applied theorem is telling me about something cool I can do with oranges.

Of course, you can also do mathematics on syntax, which is to say that we want to establish an interpretation of some of the logical/mathematical linguistic symbols as logical/mathematical linguistic objects. There is a particularly interesting domain of mathematics which is just this - Logic.

However, a legitimate criticism of our formalist programme is that it's not entirely clear whether any given theory of logic is a correct account of the cognitive resources people use when processing things using a single language module. It seems accurate to say that human linguistic understanding is messy, activating many different regions of the brain, and the simplification of language to logic doesn't necessarily give us many clues about how best to use our brains to do mathematical reasoning.

Nonetheless, it seems like a much more fruitful theory than something like Platonism, for whom the objects of mathematical reasoning and the epistemological tools we use to grasp them seem decidedly magical. "Use the language bits of our brains" is at least a potential prescription towards improving, even if we might revise our understanding of what we can do with maths more generally once we've shown that this basic bit works.

  • Thank you Paul for the exposition and I'll accept this answer. There is just one point though I'd appreciate a bit more clarity on if you have time: when you say "as logical/mathematical linguistic objects", what do you mean? Can you give an example? What are these objects, and are they part of formal mathematics or are they also practical application? Or put differently how to you get 'objects' in maths if it is just formal languages. – Neil Aug 5 at 20:04
  • Thanks Neil. One example of the interpretation of symbols as themselves mathematical objects often used is the coding of linguistic strings as integer numbers. A well known scheme in philosophy is the Godel Numbering system: plato.stanford.edu/entries/goedel-incompleteness/sup1.html However, language and syntax seem to exist in the form of words and written symbols independently of these mathematical theories; coding just gives us a simplified syntax model which we can interrogate with a bit of number theory. – Paul Ross Aug 6 at 19:52
  • As far as what the "objects" in maths are, let's take an axiomatized theory such as Dedekind-Peano Arithmetic. DPA stipulates that a particular formal relationship across members of some set, tied by a function of succession and a labelling of one member as a "zero" element. Conventionally, the natural numbers are understood as the "standard" model of this axiom system, but in fact multiple models or interpretations of the axioms. It's exactly this multiple realizability of numbers that allow for their practical use. – Paul Ross Aug 6 at 20:09
  • As a formalist, I would want to say that in as much as there is a "standard" model, it is best understood as the system of numerals "0","1","2"...and the truth of standard mathematical statements about how these numerals relate to each other via relations of equality, addition, multiplication etc. is rooted in accepted conventions of learning how to count and repeat known compositional operations. People learn how to draw the "numbers" in sequence and how to add and subtract through rote, and this sequencing is primarily what the successor function is "about". – Paul Ross Aug 6 at 20:14
  • But obviously maths outgrows its basic origins. I think it does so through the use of axioms and rules of inference stated in logically rigorous mathematical language, which can be understood in a hypothetical way; if some real world grouping of objects and relations exists satisfying the structure described by a system of axioms, then the axiom systems yield interesting conclusions we can apply to those objects and relations. Or, to put it another way, ultimately mathematics gives us useful technologies to find new ways of describing the existing complex world from simple starting points – Paul Ross Aug 6 at 20:25
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For the formalist, by definition, interpretations belong to the world of science. From a formalist perspective mathematics is not about science, it is about mathematics, which is an 'exact' science without an observable subject outside itself.

We observe that mathematics fits given situations, but those are not part of the mathematics. Geometry is not about space. Physics is about space. Mathematics is about something else, because it applies to similar situations that are divorced from space. If it were about space, why would it apply to finance?

This is not true of most contrasting positions: Platonism, Intuitionism and its related Constructivisms, Fictionalism, and other content-driven approaches.

The 'content' assumes mathematics studies something real. There is a single most honest interpretation and that interpretation is an understanding of reality, or at least our relationship to it. These can generally be phrased in terms of the connection of thought to action. For the three examples given, respectively: the nature of logic shapes reality as an integral part, the nature of logic dictates what knowledge of reality we can acquire and trust ourselves to handle properly, or the nature of logic determines the stories we believe and nothing more.

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