Mathematics is and is not formalistic

2 December, 2009 at 8:33 pm (mathematics, philosophy) (, )

Some philosophy of mathematics. And of philosophy.

The question I will cover is: Is mathematics purely formal manipulation of strings of symbols?

First, the trivial answers: Yes, since all results, definitions and theorems could in principle be reduced to such. No, because doings mathematics takes imagination, creativity, and whatever other positive-sounding things I would like to mention. (It also requires manipulation of formal systems, occasionally known as abstract nonsense or computation or calculation.)

There was this philosopher named Immanuel Kant. He proved (by very nonrigorous standards and with many hidden assumptions) a number of seemingly contradictory results called antinomies (of pure reason), such that the world is infinite and finite both. I stated something similar above, but it is fairly easy to see that I cheated, since my two statements are actually about different things.

Mathematics is formalistic in that the mathematical results: theorems, definitions and proofs – can be stated formalistically. Mathematics as in the activity that mathematicians do is not so, as it involves building and understanding (often complicated) arguments, seeing analogies, seeing past analogies, and other quite non-formalistic activities.

There is more to the question. Namely, is mathematics something discovered by or created by people? (One could ask this of natural sciences, I think.) Mathematics is discovered in that all the theorems and results are fixed, given a particular set of axioms (which means assumptions): Given proposition is true or false (or undeterminable due to Gödel’s incompleteness or even meaningless in the same sense that not every string of letters is a word or has meaning) based on the axioms in use.

Mathematics is created in that the particular results achieved, axiomatics considered, and so on, are determined by the questions mathematicians ask and answer. Physics, statistics, and various other fields of inquiry motivate plenty of mathematics, for example.

What is our conclusion? It is that the answer one finds is not important, but what philosophy does is highlight concepts that are hard to separate otherwise. This is what I find useful in philosophy. This is what I wish more philosophers recognised.

(Post inspired by this one.)


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Intensional philosophy

16 September, 2008 at 9:49 pm (philosophy) ()

This is a pretty significant paradigm shift for me. It may or may not be of personal interest to you.

Of intensionality and extensionality

First, an example from the field of set theory. If I take some set, like the set of people who read this blog, I can mean at least two things with it. The first meaning is the set of people who currently are reading this blog, or are currently readers of this blog in some other sense. I could in theory name them and list them, like so: {Tommi, ksym, Phil, Fred, …}. This is the extension of the set. On the other hand, I can mean any people who are readers now or who might be readers some day or who could potentially be readers. In this case the actual list would be largely irrelevant; the way the set is defined is what matters, not the contents it may or may not have.

A more philosophically interesting example: I have two tables in my room/apartment. Assume, for a moment, some ontological theory that ascribes existence to single material objects only, not classes of them (like “tables”). Now, consider, can I meaningfully refer to the two tables as, well, two tables, as I have been doing here? For a mathematician, the answer is “of course”. If I have two entities, I can certainly define a set to which those two belong (assuming the entities are not built so as to resist this). That: The extension of the set is what matters, the intension is irrelevant and in this case not even distinct. But a philosopher would think about the intension: By what means can I refer to the two tables, if they are ontologically only arbitrary physical objects? Certainly not as I have been doing, because given the ontology here, “table” is not very meaningful (at least obviously).

So, in closing: In math, intensionality is a means to an end or a red herring; the extensional is what matters. In philosophy, the intensional is the interesting parts, and the extensional may or may not matter. Philosophy is about the why of the world, or segments thereof, while math (and physics and other hard sciences) about the what and how.

As a disclaimer, the parts about philosophy apply most to ontology and metaphysics and ethics by Kant, and maybe to other stuff I am less familiar with. Also: Almost all distinctions are fuzzy around the edges.

What this means to me

This realisation has been fundamental in that now I no longer see significant parts of philosophy as trivial, which is useful for being motivated to actually study it.

Furthermore, I will need to re-evaluate my interest in philosophy. It requires learning a new way to think, which is generally fun and useful. Can I learn to think philosophically? Maybe. Do I want to? Maybe.

In case I am utterly wrong

In case I am misrepresenting large portions of philosophy or mathematics, please do inform me. I will presumably argue against such a claim, which will force me to sharpen my thoughts on the subject, which is useful. Or I may admit to being wrong.

In case you know something about the subject matter

I would be interested in any literature concerning this subject. Very interested. So, if you know any, I would much appreciate you sharing the information.

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Defining omnipotence

27 November, 2007 at 6:49 pm (definition, philosophy) (, , )

This is a sufficient definition and from human POV; that is, if something like what is described existed, we would call it omnipotent.

Let U be a closed universe, or something very near closed. Closed means that the things inside it can’t get to or sense the outside, and hence are unlikely to know anything about it. Let G be an undefined entity (you can read it as God if you really want to).

G is omnipotent with regards to U if G can shape U into whatever native form it could encompass. So, for example, in our universe an omnipotent G could create and remove physical objects at will, but it would not be necessary for G to be able to create things fundamentally beyond our understanding (I have a few problems with trying to create examples for certain reasons), because they are not part of our universe as is.

From this basis, a theorem: G must be outside the conception of time (or entropy or another measure of change, with apologies to everyone who knows physics for probably misusing “time” and “entropy”) that exists in U.

If this was not the case, G could first (within the dominant measure of change) create the indestructable wall and then create a cannonball that destroys everything it touches, third make them touch, which results in impossible outcome, and hence is not true. This does not happen when G is outside the conception of change as it applies in U, because then G would both create something and cancel it at the same time, which amounts to not creating the thing to start with, which leads to no paradox, because G didn’t actually create one of the conflicting absolutes after all.

The definition also assumes G is omniscient, but when talking about omnipotence, that is kinda trivial.

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The difference between philosophy and math

21 November, 2007 at 5:49 pm (mathematics, philosophy) (, )

Mathematics is the art of proving p.

Philosophy is the art of justifying p.

A proof is almost always a justification.

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A random burst of ontology and epistemology, part 1

20 November, 2007 at 7:37 pm (philosophy) (, , , )

Skepticism is a fun way to think. Global skepticism is a philosophical stance according which we can know pretty much nothing. Everyone who is not a skepticist tries, of course, to show that the skepticist is wrong. I have not seen a tight proof along those lines, as of yet. I think it is pretty futile to try, due to skepticism being right. There is precisely one thing we can know with certainty, though reasoning must be assumed to work (if it doesn’t, this entire exercise if futile, but so are any and all responses and claims of futility, so I find myself justified in assuming that reasoning does work).

The argument goes thusly: I think, therefore a thought exists.

Do note that “I” does not necessarily exist. Or the thought might be momentary; time and other measures of change may be illusions.

The skeptic can’t really touch that argument. Neither is it particularly strong argument. The existence of a world in which we live would be nice to know, for example, and would be a lot stronger claim. I just can’t figure out a way to prove it without nontrivial assumptions. This is why I will assume it and the capability of know things about it. There are further justifications for that assumption, gratefully.

It is a fact that I perceive something around me. I will call the immediate source (as opposed to the ultimate or final source, if there is such) of these perceptions a world or a reality or some word that is practically synonym thereof.

If there actually is no world, I will lose little by assuming it, because I can’t perceive whatever else there may exist (otherwise it would, by definition, be part of world, which is a contradiction). If there exists a totally irrational and random world (defined as one about which useful knowledge can’t be gained), I likewise lose little, because no matter what I assume or don’t assume, there is nothing useful I can know about it. At least trying to find patterns keeps me well amused. If a world about which something useful can be known exists, it is smart to assume so, because it is true. A world which works so that all human assumptions about it are false is contradictory, because one could assume that all human assumptions about it all false, which leads a and not a, where “a” means “all human assumptions about it are false”.

Hence, it is justified to assume that there is a world about which one can know things.

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Comments on a bachelor’s thesis on Gadamer and roleplay

20 November, 2007 at 5:58 pm (academic rpg theory, philosophy) (, , , , , )

Opusinsania wrote a bit something about Gadamer’s understanding of play and roleplaying. It is in Finnish, though. He may be willing to share the writing if asked, or not.

A very brief summary: First part of the thesis is about Gadamer’s philosophy and such. The interested should utilise Google, Wikipedia and the local university for additional and actually accurate information. What follows is my on-the-fly translation and commentary of the relevant bits. You have been warned.

Gadamer consider both play and art to be fundamentally the same thing, at least in that both of them interact with the person experiencing them and the interaction becomes primary, in a way, and separate from the rest of the world. I am reminded of Christopher I. Lehrich‘s article Ritual Discourse in Role-Playing Games, hosted on the Forge. Back to Gadamer. In his philosophy, a play (or game or art) attains perfection when it becomes a structure, or form. In theater this happens when an actor stops acting and becomes what she was trying to act. A similarity with (some definitions of) immersion and gaming are clear. Gadamer further says that the players will focus solely on the content of the game, which reminds me of flow, which, consequently, has sometimes been connected with immersion (discussion on some years back). In spite of this, play can’t happen without a player.

Turku school manifesto by Mike Pohjola was quoted: “Role-playing is immersion (“eläytyminen”) to an outside consciousness (“a character”) and interacting with its surroundings.” A definition of immersion by Pohjola: A process in which the actor’s consciousness melds totally into the character’s consciousness. This is seen as the goal of play by Turku school. Pohjola has said that perfect immersion is impossible to attain, but can be approached (see also: limes in math). One way of understanding this is to think that the player’s consciousness is negated. Pohjola has not shown public support for the alternative (that immersion is a relation to another ontologically independent entity). A third alternative, not from Pohjola: That roleplaying is taking a social mask, stolen from ludology, and immersion has little role. There are intermediate positions.

Another important concept is diegesis, defined as everything that is true within the game. If you are into Forge theory, shared imagined space (SIS) is not that different a concept.

The purpose of gaming can be defined as immersion, the simple joy of gaming (or flow) or social reasons and other concerns outside the game. Aside from Pohjola, every definition of rp considers the social dimension to be essential.

The meat of the thesis (IMO, at least) is the contrasting of Pohjola’s and Gadamer’s understanding of roleplay (play in Gadamer’s case). Gadamer requires interaction to call something play. This can be interaction with, say, a deck of cards (solitaire) or other players. The degree of interaction required is unclear. This can be seen as supporting Pohjola in that one can play by himself (I certainly do), but it is still possible to define roleplaying so that it requires social interaction (which I find smart). Some pervasive games (which are at best fuzzily separate from normal world) may complicate the matters a bit, but I am not sufficiently interested in them to say anything further and they were not mentioned in the thesis.

Both Pohjola and Gadamer consider play to take precedence over normal life and create some sort of self-contained reality. Pohjola took the concept from Hakim Bey.

According to the thesis, a significant difference between Pohjola and Gadamer is related to Gadamer’s claim that game becomes primary and player secondary, while in Pohjola’s model the player consciously becomes the character and then the immersion happens within the player, so the player is still central. In Gadamer’s model the player no longer exists when the games achieves perfection.

I disagree with the point (badly) referenced above. In both models, the player disappears. In Gadamer’s this happens as the game reaches perfection, but the player is still necessary, as there can be no play without someone playing, as was claimed earlier in the thesis. In Pohjola’s model, the player disappears as he achieves perfect immersion (since this is impossible, mere sufficient approximity to this stage can be assumed, but it doesn’t change things significantly, IMO), but is still necessary, due to the other consciousness being the same actual person. Or, if this is not the case, the game would partially happen outside the observable reality of humans, and is thus quite irrelevant. Essentially, I disagree with the point that there is a fundamental difference between those two points of view, but can’t actually say why, probably due to not understanding the argument well enough and due to it being hard to prove negatives.

Opusinsania, if you don’t want this text to be public or here at all, say so and I will act accordingly. If you do decide to publish the paper somewhere, I’d suggest The International Journal of Role Playing.

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