Just finished listening to the discussion with Victor. Here's my contribution (with a couple of my personal comments in square brackets) in the form of a partial transcript (you guys planning on providing same)? Worth putting in typeface, I reckon.
Victor: Given the assumptions of the Peano axioms of number theory, given the established meaning of the term, 2 as the number that follows 1 and 1 as the number that follows 0, it is possible but purely syntactic manipulation to arrive at the conclusion that 2+2=4. But, that--but, you see, the difference here is that when we talk about Euclidean geometry, geometry is something that is empirical. Itâ€™s supposed to apply to the real world. There are actually shapes and lines and figures and angles and, as long as you relate geometry to the real world, this is what happens. People start out with axioms that they didnâ€™t properly support, they cannot contradict it, and then eventually they discover that this axiom is not actually necessarily true. That is not the same thing as what is happening in 2+2=4. 2+2=4 is true strictly by virtue of the definitions of the terms. But then, that is all the term means. What we may discover in the future is that the mapping between the concepts and the empirical objects isnâ€™t the same. In fact, we discovered that already. If you mix, for example, a litre of water and a litre of alcohol you will not get a 2 litre of vodka, you will get slightly less [laughs--Ukrainians!] because when water and alcohol mix they are not linearly adaptive. So, if you attempt to relate this to the empirical world, we have already proven it false. But as a purely analytic statement--I mean, itâ€™s not even, strictly speaking, analytic; itâ€™s simply syntactic manipulation. You are just pushing symbols around according to given rules. In terms of definition, in terms of pushing symbols around by the given rules, 2+2 is certain. But it is only certain as long as it doesnâ€™t actually mean anything. Thatâ€™s my point. That you can either push symbols around and have your certainty or you can have--you can introduce--you can relate it to the truth about the world and say something useful and lose the certainty. You canâ€™t have both.
David: Well, I would disagree with that. There are cases where you can have the best of both there and, for example, if we take the concept of a thing, which I define to be some portion within the whole of nature--the totality--so, a thing is something that is limited in extent then we can conclude that everything within the world, all phenomena--clouds, galaxies, people, thoughts, electrons, models, whatever--theyâ€™re classified as things so there is an empirical correlation right there and then whatever we conclude about things through a purely logical process applies necessarily to what is in the world. For example, we can conclude that a cloud is not the totality of all there is, itâ€™s a thing within the totality and so there is a definite truth about an empirical phenomenon thatâ€™s arrived through pure logic.
Victor: So, but you see, again, the conclusion you make is really a purely tautological statement and the statement is something that is less than the totality is less than the totality--
Victor: But, now, try saying something useful about the cloud. Try saying that a cloud is not infinite.
David: Well, a cloudâ€¦
Victor: Thatâ€™s right, thatâ€™s right.
David: Yeah, a cloud is not infinite in the sense that itâ€™s existence does not extend indefinitelyâ€¦
David: It has a beginning and an end.
Victor: But there are things which are potentially infinite, empirically. Black holes, for example. Black holes have certain characteristics which are infiniteâ€¦
David: Well, weâ€™re talking aboutâ€¦
Victor: My point here is that in making these arguments, youâ€™re not really saying anything about the things. Youâ€™re saying that which is less than the totality is less than the totalityâ€¦
David: Well, we can say certain things about it.
Victor: You should try saying something useful. Like I said, for example, â€œAll things are finite.â€ If I recall correctly, that is actually one of the conclusions you have tried to draw a long time ago--a few years ago. That all things are finite. And I just gave example of how the moment you try to relate your purely syntactic games to the actual, empirical concept, it immediately falls flat.
David: Well, no--no. As long as you are clear about what finite means. Soâ€¦
Victor: Exactly. And then what you are saying is because finite is something that you define as having limits, being less than the totality, then if you use your own custom definition of infinity, then all you are saying is still something that is less than the totality is less than the totality.
David: Yes, OK.
Victor: As long as you are using custom definitions, you can have your conclusions. But they donâ€™t actually mean anything because all they are is your conclusion about your own definitions. You arenâ€™t really saying anything about the thing.
David: Well, OK.
Victor: Youâ€™re just restating your own definitions in different ways.
David: If we--I understand your point--if we just focus purely on the form--yes, I agree itâ€™s a tautology, and focussing purely on the form, a tautology has no meaning. Itâ€™s just a re-statement.
Victor: Not quite, not quite.
David: If we factor in the content, then it becomes meaningful to usâ€¦
Victor: But thatâ€™s my point.
David: â€¦as living human beings. So itâ€™s the content
Victor: Itâ€™s the content.
David: â€¦which makes the tautology meaningful.
Victor: A tautology, generally speaking, can be actually very meaningful because--think about it this way; tautology explicates the implicit. In mathematics, for example, any mathematical theorem is, strictly speaking, a tautology. But there are things that are implicit in your assumptions without being obvious and it takes a theorem, which is a series of tautological statements, to explicate it. So a tautology in and of itself is not a bad thing. Tautologies can be very useful.
Victor: The real problem here is that when you try to relate your tautology to the world you immediately lose the certitude that you are seeking [Exactly! Good point--frankly, I think these guys are in agreement, even though they appear not to be--at least to one participant?].
David: Alright, a change of subjectâ€¦[end of voluntary partial transcription service--that I sure hope has not been already provided elsewhere]