Philosophy of mathematics.

[ferrus]

Where do you stand? Intuitionist, formalist or Platonist? Why?

Reading material:
https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html
http://www.ams.org/journals/bull/200...99-00802-2.pdf
http://people.ucalgary.ca/~rzach/static/hptn.pdf
http://scienceblogs.com/goodmath/200...and-semantics/
http://www-history.mcs.st-and.ac.uk/...Intuition.html
http://plato.stanford.edu/entries/ph...y-mathematics/

[Sappho]

An inferior knowledge of geometry disqualifies me from passing a notable verdict. Kant, however, does have convincing arguments for the a priori conception of space and time as well as the limitations of perception itself (with obvious consequences for mathematics), and Schopenhauer develops this nicely and elegantly into his principium rationis sufficientis essendi. - Still, at this point I have read too little about 20th century mathematics to state any relevant opinion with conviction.

[ferrus]

Brouwer mentions Kant's ideas - which interestingly gave Gauss cause to ponder whether he wanted to tangle with Kant on the issue when he studied this matter and let Lobachevsky reveal the first non-Euclidean geometry - which neatly segue with what I consider the local intuitionism of basic mathematics:

But the most serious blow for the Kantian theory was the discovery of noneuclidean geometry, a consistent theory developed from a set of axioms differing from that of elementary geometry only in this respect that the parallel axiom was replaced by its negative. For this showed that the phenomena usually described in the language of elementary geometry may be described with equal exactness, though frequently less compactly in the language of non-euclidean geometry; hence it is not only impossible to hold that the space of our experience has the properties of elementary geometry but it has no significance to ask for the geometry which would be true for the space of our experience. It is true that elementary geometry is better suited than any other to the description of the laws of kinematics of rigid bodies and hence of a large number of natural phenomena, but with some patience it would be possible to make objects for which the kinematics would be more easily interpretable in terms of non-euclidean than in terms of euclidean geometry.

However weak the position of intuitionism seemed to be after this period of mathematical development, it has recovered by abandoning Kant’s apriority of space but adhering the more resolutely to the apriority of time. This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness. This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal number ω. Finally this basal intuition of mathematics, in which the connected and the separate, the continuous and the discrete are united, gives rise immediately to the intuition of the linear continuum, i. e., of the “between,” which is not exhaustible by the interposition of new units and which therefore can never be thought of as a mere collection of units

I mean, I do wonder to what extent one can actually say the other four axioms of Euclidean geometry are an intuitionist construction? Even the most simple models of geometry rely upon lines and points as much as all basic mathematics relies on intuitions of numbers. In some sense for us to even develop these concepts it was necessary for us to have evolved intuitions consistent with physical cause and effect and basic mathematical properties of the universe around us - much as basic recursive grammar is also to some extent an intuition upon which we put a model.

The question I'd rather want to ask is this: even if intuitions form the start of mathematics, are they in turn mathematics? We have intuitions about space and time which got refined from Aristotlean models of kinematics, to Newtonian, to General Relativity. Each step required turning away from intuitionist thinking and embracing a formalism which allowed itself to speak its own language. There is always an intuistic residue - but this doesn't justify its continued existence. They are rules which we are pre-programmed to accept that, in the due course of elaboration must either be validated or eschewed depending on the need of a deeper model, they are a part of our human cognitive capacities, and are necessary for that, but they are still no more important, or less than a formal axiom derived through non-intuitionistic means.

[Zephyrus]

I regard any realist position on the matter to be as crazy as dualism, so I am an anti-realist. However, I can't be more specific than that because I am not going to look in to the arguments for and against individual anti-realist philosophies (e.g. formalism and fictionalism). I am done doing serious analytic philosophy.

[P-O]

I guess i'm a formalist. But i feel like if i understood it better, i could be convinced of the idea of intuitionism.

Formalism is true; one doesn't have to have a conception of a particular physical reality in order to do mathematics. The axioms define a system and you can unravel it without thinking about anything. In practice, however, the intuitionist notion rings true. The purpose of mathematics is to give a structure to our ideas. It's a tool that maps concepts onto a language.

It's not exactly clear to me that they're actually mutually exclusive in their most general sense even though they appear to be diametrically opposed.

[Utisz]

Reminds me of these videos, which are quite accessible as an intro to the topic:

[Hephaestus]

I guess i'm a formalist. But i feel like if i understood it better, i could be convinced of the idea of intuitionism.

Formalism is true; one doesn't have to have a conception of a particular physical reality in order to do mathematics. The axioms define a system and you can unravel it without thinking about anything. In practice, however, the intuitionist notion rings true. The purpose of mathematics is to give a structure to our ideas. It's a tool that maps concepts onto a language.

It's not exactly clear to me that they're actually mutually exclusive in their most general sense even though they appear to be diametrically opposed.

I'm probably talking out my ass here, but I don't see any contradiction between formalism and intuitionism if you consider the application you describe here. The 'real world' observations are just a source of axioms to define your system.

Formalism for the win. The others are overly constricting (when not bloody wrong).

[JohnClay]

I think it is all about patterns and symbols themselves involve patterns. Manipulating patterns happens on a mechanical physical level like a machine. What category does that fall under? (Sorry I don't understand this discussion very well)

I think my philosophy is physicalism. I think that means I'm an anti-realist.
http://languagelifeandlogic.blogspot...ems-for_7.html

There are, of course, many competing philosophies of mathematics, some of them (like platonism) realist (in the sense of accepting the real existence of abstract mathematical objects), others anti-realist. In general, the former approaches seem more or less incompatible with physicalism, and the latter compatible.

http://philsci-archive.pitt.edu/8847/1/physform-csf.pdf

I voted for formalism though there might have been other options that are also physicalist.