Frege, Russell & F.o.M

0:00 Thank you. I'll try to speak clearly in English, but if you have any problems with my English answers, you can just stop me now and repeat. And also, since we're fairly small, if people have questions during the course of the presentation, please do interrupt and I can say something before going on. Well, the topic that I'm going to address is the relationship between... Anthroposophy is concerned with the foundations of mathematics. And in particular, some of you may have heard of Frege. And Frege is involved in the history of anthroposophy. So clearly Frege, and instead Russell as well, are crucial in the history of anthroposophy. And crucial, I would argue, is to be the main lead of my book. Because of their interest in the foundations of mathematics, I think it's still certainly crucial. In my view though, it is only one part of an interesting concept, an academic conception, that comes through Frege and Russell. It's to some extent a complete elipso in the context of the world. Whether or not one can believe it or not, it's either actually concerned with mathematics or developed in mathematics.

2:30 Mathematics, I've already talked about mathematics, but mathematics I want to be good at. So, if you please, take it a little bit slowly, because it's possibly because of the room also. No, you're quite right. I often say, before I give a talk, as I do, as an echo to it, would it be better if I spoke? Would it be better if I stood up? Is it better if I stand up?

5:00 I don't know. Just a very short question. When you say something like early modern philosophy is inspired by analytic geometry, do you really, I would say, think and insist that it's kind of a one-way thing, because in some perspective we can just put it the other way around, right? No, no, this is just a very broad definition. Yeah, I mean, I think the divisions would be, to the extent that they have some, and they certainly have, but one just does. To the extent that one does that, one of the main explanations of that is that something new hasn't occurred, and if one is going to identify something, the concern is that this is, as I say, what analytics really means. To one of my philosophical or physiological questions, to misguide a protein with something, an issue can be given, can be displayed or reconstructed.

7:30 Most generically, people can give a definition. I distinguish three main modes, I call them modes of analysis.

10:00 So, they're not things that haven't worked, but it's worth distinguishing that some of them are working. So, these are the regressions. We work back to something like this. They immediately think that when you look at what philosophers are actually doing, decomposition is only one part. So, decompositional concepts identify the elements that are really part of them. They're particularly characteristic in the frame of Russell's question. What was once on record is that I think it's tended to be overlooked because one thinks of analysis May I ask a question? Frankly, probably I'm inclined to write or to think about analysis and decomposition, but I think there could be some reasons for that, because, say, if you're just giving this very general definition, right, working back to principles, then suppose that... Like in Berkeley, you think about principles as somehow synthetic, so you would need to rather put things together to get to principles, and that probably wouldn't be analogous, or it still would. Well, you see, there is a comeback by the different positions. You might say that what someone's doing when they're trying to work back to principles is actually taking the system as a whole in some kind of sense.

12:30 In geometry, axioms of geometry was totally reversed not only in ancient geometry. In the beginning of the 20th century it was very strong, the urge of Hilbert was very regressive, I think even more important than ancient geometry. I don't agree. What I want to say is that these conceptions are still alive today. All I'm saying is that this conception was dominant in the ancient period. Official conceptions in analytic philosophy may be that decomposition analysis, actually when you look below the surface, as it were, they're actually using algorithms of analysis. So a good example of this is... In regard to Russell, he wrote a paper in 1907 on the regressive method of discovering the premises of mathematics, this conception of analysis. He doesn't hold it in that way. In, for example, theory, this analysis is decomposition. But here we have a paper in using analysis in this regressive sense. So it has a long history, but it's still there, uncertain. But what I want to say is, if you like, one thing that struck me is just what a discrepancy there is What they do and what they say there, they'll often say there, using a certain method, not entirely clear-imagined, so there's a certain tension between the results and, again, once again, the fantasy to re-evaluate.

15:00 Decompositional work, although that, too, is involved in ancient Greek geometry, and if you've got an interpreter who reads this book on Greek geometry, they, in a sense, try to give it a go at it. I quite agree with you about Descartes and Kant. I think that their notion of analysis is just not what you want. But I think that Leibniz's notion of analysis is just not what you want. There are much more regressive communities with that, but Witten doesn't. Good. Yeah, I think there are regressive conceptions in Leibniz as well. I think what you could say occurs in Leibniz is a kind of matching of a regressive conception and a decompositional conception. So, a data with decompositional conception, insofar as it relates to conceptual analysis, is Leibniz.

17:30 The attainment principle here, where in every internal composition, let's say contingent, universal, or singular, the notion of the creditors is obtained in some way or another by the credit card that connects to the agreement is transformed into what you find in a particular learning form. A synthesis of concepts is seen as introducing a new form of a request for conception as this works-back synthesis. But sometimes, maybe I'll say something about it, which says, listen, this interpretation is needed in order for the resources.

20:00 It's an art of analysis. Is it stressful to you? I really cannot believe it. It's totally the country. Algebra is analysis. It's not at all of analysis. I haven't got this paper. Let me just explain. This is false. You have to build calculations. It's not analysis. You just decompose the things. You have to build them. No, no, this isn't the problem. What I've got to say is, this moment of interpretative analysis comes to the fore as a conceptualizer of an essay as such, and it only happens to be possible. I'll say more about it in a minute. But the basic idea here can be found right from Richard Wynne, John Jones. So if you think in geometry, for example, you could say, and here's a quote from Coppess where he identifies six stages in geometry. The first two say enunciation for classes, state what's given and what's been sought from it, for a perfect enunciation for the supposed parts. The exposition, their thesis, takes separately what is given as an advance to the use of the presentation. That's before one starts to work at what I'm figuring out about new constructions. So this is what I would like to say is an interpretation. You compose a little bit, you enunciate something, you interpret it and so forth. So one can find that idea in geometry. Most importantly, what science and technology do, like geometrical problems, are translated into the language of arithmetic and algebra.

22:30 So when I talk about algebra and art of analysis, I mean in the 16th century, in the end, the algebra was called art of analysis. And the question is, in what sense does it matter? In what sense does it matter? Well, what I need to illustrate with this is that it involves transformation and translation into some particular conceptual framework. In this case, the point being, this issue far richer results to solve the problem than what you've done to your translation. What I want to say is that this process of transformation and translation is part and has to be recognized as part of analysis. And you don't have the answers. You cannot do it in analysis. When you only translate the problems, the questions that have been solved yet, you cannot complete an art of analysis. You just translate problems, for instance, many problems in mathematics, you just translate them and you make up 30 different formulations of the four colloquial theorems, and no 30 different probes, only one probe, but 30 different formulations of the same problem. So, either in algebra, using shift theory, or using many different languages of mathematics. So I'm saying that the problem is not enough, it's something like a solution, which is much more important. I agree. What I'm saying is that this interpretive analysis is one part of the whole process of the analysis, but this part actually gets thematized, conceptualized, to a much greater extent, if I'm not wrong, than ever before. It's the reflection on that process that actually drives the energy. I think there are at least two different related processes of the analytic geometry and one thing it's something say common for 19th century whereby analytic geometry is normally kind of arithmetic

25:00 Translation, putting your words into arithmetic equations, or if you go back to the 17th century, it's not actually the case, like in the sense of Descartes it would be analytic geometries. Much more closer to algebra, something closer to today's algebraic theory. I think it is better that we leave it here and discuss it. But that's probably a better question. So when you say analogy, geometry... I'm happy to learn through different senses, for example. Okay, so let me do... is between the translation or interpretation, as I want to say, of geometrical problems into the language of algebra, and the translation of the propositions into the configuration of the credit card. That's why I want to say that Frege's certificate lies in the dynamic acoustic, through iteration of it. And the semantic question that that opens up, and his reflection on that, and so on. That's kind of absolutely fundamental, not the core element in the academic literature. So here's some examples, and then I'll have a minute to elaborate on the details of Frege, so I can say more about his recipe as we go through. Frege's absolutely core technique in the Grunwald and Auerbach method is that the number statement contains the assertion about the concept. So here's the example. Jupiter has a problem. You can see the problem. Just treat this as a naive decomposition of the subject code. That's the subject code. What Frege said is that should be interpreted as this, namely the concept that the moon of Jupiter has four instances.

27:30 Of course the void being that it has four instances is essentially a default for the biological divide. So he's saying that when we say Jupiter has four instances, we're saying not something about it, but of a concept. So this is what I want to say is interpretation. It's paraphrase. It's translation. It's transformation. Part of the point being, we've already formalized this, but there's also some philosophies. It's saying there are standardists there to see what's really going on. We're really talking about, when we make analysis, we're really talking about concepts and saying something about concepts. So, against standardists, we take cases of negative existential statements. To which, according to Frager, the state is involving the number zero, and when we say that something doesn't exist, we say that the number attaches to the relevant concept of zero. So what we say is that unicorns do not exist. Again, this is standing in debates between them, debates about what we buy and what we sell and so on and so on. What are these unicorns, these objects that have this property and are not existing? The definition is how can they be anything? They can be subsystems. They can be something that has nothing to do with humans. We haven't got to suppose subsystems. What we're doing is we're talking about the concept. When we say unicorns, we mean the concept is not extantial. When you say what you mean, what is you? What is we? When we say unicorns do not exist, we mean the concept of unicorns is not intangible. What is we? What is important for me. Okay, good. So that's a major question. If one sort of bit continually climbs, there will be certain philosophers, perhaps like Michael, who might be tempted to ask questions about what these things are.

30:00 And then, if we correctly understand what's really going on when we make such claims, we recognize that we're really talking about concepts. We know what a thought is, we know what a form is, so we can frame for ourselves the concept of a uniform. I think that if you say we, you mean what a French person would say when they use the third person singular in a personal construction. Au revoir. We make the difference in English, usually in text, often we make the difference, and we are the impersonal formulation. So if I say one, one might be tempted to think, in using words of the universe that do not exist, that one might be tempted to ask, what are these unicorns that have this property? And when one recognises exactly what's going on, one recognises that impulsiveness... The purpose of this is to illustrate this distinction between compositional analysis and interpretive or paraphrasal analysis. The right way to analyse that is not to break these parts, given by a superficial, in a different form. What's really going on is these interesting. So, it has to be these ones, not only compositionally, but paraphrasal. And, you know, in case of very vivid drawing, it has to be a logical one. Sometimes one says that it's instinctual.

32:30 And you're saying that the right way to see it is in terms of concepts being instinctual. Yes, something stronger. Whatever can be quantified in addition. I think it's not topology.

35:00 Well, I think you're right. And this opens up questions about the legitimacy of this. As soon as you've got this... The question of, well, is it right? Does this mean the same as that? And you want to say, well, in one sense a mean, yes, in another sense, no. And then you have to say, well, what are the concepts of the meaning under which it is and it isn't? That's exactly what... Questions that have been driving the analytic process from Frege to Rasmus. Okay, we should go on for a little bit. So, okay, this is also, this is in Frege, an important contrast which I think is not sufficiently recognized between Frege and Russell. Actually, their philosophical perspective is quite different. This is very much a risk to my meal, the idea of there being interpreter frameworks in which you are translating the philosophical problems. If their interpreter frameworks are different, they're actually going to get different results. So you won't expect Frege and Russell, even though they're both logicists, to have the same answer to the question. So here's an illustration. Frege's famous problem, one of the most famous, the so-called power of... What I'm using is to show that in the theory of descriptions, Russell quite explicitly uses it for what you eliminate the need for which the problematic definite description disappears. In the correct formulation of presently in France, there's one and only one. There is no phrase. Definite description. So it's used as though, as I suggested, statements of numbers in this problem which I'm just going to.

37:30 If the concept is saturated, it must refer to a saturated object. The concept force is an object. Since objects are non-concepts, the concept force denotes an object as a non-concept, so that's false. So this is the puzzle. Frager says that what we read in the description is the extension of the concept. So it's still called that, it's false. But this is a paradox. How might one solve this paradox? Everything is either or is something that to those that fall out of it so to say that something is so to talk about is to say that there's something that some objects are and all the other objects are so to say that something is to say that something is a wall is to say that everything is either or or something to do partition or we do have a concept that you know makes use of the day. I'm out. What's that? What's a real number?

40:00 It's the partition of the ration. Oh, that's the idea, right. It was just the way you phrased it made me think of that. Yes, but it's possible there's some connection there. But I think the general idea of your initial first thought was just to take the sentence, this is the surface of mathematics, and suppose that the criteria for something being an object is whether the name for it would form the F. Form the F would denote an object. So I think it's probably more general, I'm afraid you insisted the concept had to be sharp, had to be defined for all objects, and that's something again you find in the maybe dimension.

42:30 Did you want to...? Well, just a nice question. A differentiable manifold is a manifold. The concept. But then the second, the translation of Frege is something much stronger, something that the principle excludes in meaning, which is much more... Oh, I agree, I agree. As I say, I'm not saying this is the right answer, what I'm saying is that the possibility was there for Frege, given this deception of concept, for which the law of exclusion has shrunk, to offer a pair of phrases, an interpretation, which would solve the problem. But what's interesting is he doesn't. There is still a certain tension between his use of interiors and his use of decompositional amounts. I've got some other intelligence on the subject, directly to do with the subjection of the 9500 question. Actually, a related thing. Of course, if you say something like all objects, it's obviously problematic. But I just want to use it really explicitly for the idea that we should... Think of something like, oh boy, it's already in microdynamics. Yeah, absolutely. You just thought there was a domain of all objects. No, that's a word type description. You know, there's objects, extensions of concepts are also objects. That's all I'll come to in a minute. Okay, and concepts are simply partitions, if you like, of the domain of objects. So basically, for any object, you've got two to the n concepts, and that's it. They're extensionally defined, and there's two to the n possible partitions, so you've got that many. And there's extensions of the concepts which then somehow also apply to the debate. You immediately see that the constitution is going to arise from that conception. So it might end up for the main debate. Okay, so let's move on to the next thing. So we've had some examples. It's to do with, if you like, it enables us to offer a diagram, conceptualize what's gone on. So the semantic issue is the role of abstraction.

45:00 So in, because of the role of abstraction, people like Rickards are called To say that two concepts are equinumerous is to say that the number of F's, interestingly, can be used to derive the question that comes to the stages of that. Logic is to see that there is something still there that works, or a phrase is there that is still going on.