Timeline for Does mathematical formalism have an opinion on semantics?
Current License: CC BY-SA 4.0
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| Aug 6, 2019 at 20:25 | comment | added | Paul Ross | 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 | |
| Aug 6, 2019 at 20:14 | comment | added | Paul Ross | 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". | |
| Aug 6, 2019 at 20:09 | comment | added | Paul Ross | 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. | |
| Aug 6, 2019 at 19:52 | comment | added | Paul Ross | 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. | |
| Aug 5, 2019 at 20:04 | vote | accept | Neil | ||
| Aug 5, 2019 at 20:04 | comment | added | Neil | 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. | |
| Aug 5, 2019 at 18:04 | history | answered | Paul Ross | CC BY-SA 4.0 |