Natural Numbers / Frege & Peano / Mathematical Forms (contd.)

0:00 Okay, so, the final session by Charlotte Herndel from the University of Pennington Bridge, and she's talking now on this formation of justification of science. It is shredded. It is shredded. It is shredded. I don't know. I don't know. I don't know. I don't know. Thank you. Yeah, I don't want to thank you for coming. I apologize for changing this title. I'm going to do it again. Too much general motivation. I mean, if you look at mathematical practice, then you find that how definitions are, and definitions are clear in the literature. for a definition to the work that you have to do with a few reasons, or a case model specifically. And it's often also said that we can hear that arriving at the risk of a definition is a more representative in mathematical knowledge. And the initial reason for a definition often will be used as how the mathematician itself First, so in which ways are definitions justified and are those kinds of justifications and these reasonings? And the second question, in which ways is the definition of the definition of the guide and are those kinds of organizations? And it's important here, what I should say, that I'm only really concerned with the guidance of formation and not any other aspect of the organization. because they obviously may not ask you about the definition. And I think while it can be, I mean, in principle, the justification information are actually very different things, but I think if you look in mathematics, how it's usually done, it's clear that the initial justification, or the first justification for a definition, this usually shows you and tells you how the information information of the definition was guided. So this means that it was in both photos that I don't know was to have to say information and justification. I will only speak about all kinds of justification

2:30 because this corresponds to all that kind of . So just a general question now to a big outline. What I will first do is to main, I think, account in the literature on the kind of things, as it applies to actual practices. So we'll speak about Lackagos' account of two different definitions. I think it's extremely important, this account, but I think it's also incomplete and also more important and so on-sided. So we'll, in order to say what I mean, I will consider a case study of notions of chaos and then identify the other kinds of communication and relations. And in the relationships between them, we'll speak more explicitly of the implications for Lackawas, the power of culturation and summarizing ourselves. Okay, then let us go to these Lackawas, the power of culturation definitions. So his main examples in his main books where he introduces this proofs and refutation space is the evolution of definitions of polyhedron to definitions that really ensure that you can prove the specific conjecture that the number, all the conjecture ones, sometimes the number of workers in science, the number of edges plus the number of faces, the definition really shaped this kind of conjecture. And unfortunately, the definitions gives many examples, and it doesn't really use the concept of definition. And it's clear that, as the first point, it cannot mean that mathematically definitions are generated to be eventually . It's quite trivial. I think this is with every definition, and every definition is proof-generated. Okay, and then what I think best factorizes to be able to find in Lapidus' writings is to say that a proof-generated definition is a definition which is only formed in order to be able, in order to prove, in order to be able to prove a specific conjecture which is regarded as value. Now, it sticks to all of that of those examples, and it also fits well related to what he really says about truth-generated definitions. And there's one point here which I should further mention, because Lagardeau always discusses this truth-generated definition in the context of this method of proof-sumplification

5:00 which is mainly on this method. So one good thing should I define proof-generated definitions in reference to this method of of proofs and reflications, but I think that's not a good idea, simply because some details of this method of proofs and reflications are simply not relevant to, in a way, how definitions are formed and justified. For instance, this has reasonably been argued that on a straight forward strict reading of this method of proofs and reflications, it has to be wrong because it really exaggerates the role of how it is. Now, even if this were, so it wouldn't be difficult, like the poem, maybe some definitions of true legal, and another example by Lafteroschweig is part of the original definition of nation. Here, good, and then Lafteroschweig also suggests that true definition is also true justified, and he mainly speaks about generation and not about justification, but it's implicit, I think, And if you would ask him this, then he would say that that's the justification for proof-generated definitions. Are proof justified, meaning that they're needed in order to be able to prove a specific conjecture that as well? And yeah, I want to make a more trivial point. If the conjecture is mathematical, then he would say that proof-justification, also proof-generations is a kind of reasonable kind of justification formation. And Dr. Josh does definitely think that this kind of formation is reasonable. of this book, what does Vatavich just say on the appearance of true degenerations? In his writing, he portrays it with his examples and case studies as really the best way to degenerate definitions. And he, but he does press in his PhD thesis and only in his PhD, in his PhD, in his PhD, in his PhD. He did not advocate that all definitions of mathematics are, but should be proven because it's just things he has identified one of his covering. There are simply other kinds of notions as well. But I think, I mean, he is committed from the time to really characterize the theories as The way in which definitions are formed and justified. right. He is indicating that, when you look at mathematical fields, that these fields

7:30 for two generations were in a so important way, a part of the conditions that they were justified or not exceptional. And just as a final very quick point, Bakadou also does claim, because I mean, he does claim in this book kind of way that he's more conformed, is more of a concern with informal mathematics. And he has doubts that in highly informal mathematics theories, his thoughts apply that he does say that it applies to all the different mathematics. He mentions it's physically metamorphic mathematics, and he mentions it's really safe . And I think this Dr. Dostakhan is very important, but I think there's more to say, and I do some training for his case study. And the case is a regular behavior showing kind of sensitive things, as well as really more, and the idea that small errors in initial conditions lead to significant outcomes. And what in a random way of a system that involves a theory, which is the theory of the theory of dynamical systems stretches from the third is where we're able to hear it was, you know, amended to that present. Why notions of tears, very quickly, this is really, I think there are two reasons, first the picture is different, and it's different, I think it's not the exception of other things, but I think the picture is very, in a very wide spot, I would say more about that. And second, I think, I won't, I mean, I won't say more about the second point here, because we don't have time, but there is itself a very important notion if we, partly of how I was really coming to this topic is, or arriving to treating this topic is really, if you want to understand the chaos, if you want them, and there's a confusing variety of notions of chaos in the literature and randomness and things like this, and if you want it, I think it's in some cases, at least which problems, then the kind of justification of which way they are justified, So I want to say now, on the basis of the case studies, the other kind of justification from rationality. First of all, condition justified. And here you have the definition of justified, but the fact that it's equivalent in a natural way to previously specified conditions,

10:00 regarded as mathematical. It's not a very abstract definition, let's have an example. There were very, very many examples in my case study. Examples with mixing. So there were two, and we're totally independent questions. Let's first have a look at the first. We start with the notion of erudicity is, especially in the 1930s, it's still today a very important notion because it's believed, and still believed can be somehow very important for securing the foundation, And so you, there's the question when you have a single system and then you consider, which is a natural, just two copies of the single system and you want to go, when this composite system, existing of these two single systems, what exactly, which kind of property does the single system have to have, that exactly this is fulfilled. It's the first question, and then the second question. There's a spectral theoretic account of the network system theory and the Morsev theoretic account, roughly speaking. And they are linked, and there are many of which theories. And you find that in the spectral theoretic account, there's a very important notion of having a continuous spectrum, which roughly means that when you look at the another system that is simply the simple value and the only value. And then you just asked with which such a property corresponds to this and continuous spectrum. And the answer in both cases is this definition of . I won't really go into this definition for a time, but if there are any questions later, I'm happy to discuss it. This is just the P.S.L. So, these are some comments on how Martin's first book will learn about, in one of the first books on ergodic theory, saying that weak mixing is not just in the loop in partiality, it comes from the fact that it's equivalent to some rather natural geometric and functional conditions named exactly what the system is ergodic and consumer spectrum. And this is The code is representative in the farming period later, up to today, some of the main justifications for it's missing. And then two of you comments about justifications. And first, I think there's a little danger for a specific definition,

12:30 not necessarily for all, but for specific definitions. If you don't see that they're what I call condition justified, and you think there has to be some kind of intuitive meaning, and you try hard and say, I want to understand the definition. that has to be an intuitive meaning, but we try hard and eventually arrive at one, but it still seems odd. And I think they have to be done completely . Some of the different notions were interpreted intuitively, but I think this doesn't really make sense, because that's not the way they are, and we've never justified. And this is very similar when the English language speaks, And the danger of interpreting was, of course, proof-generated and proof-justified definition is intuitive because this may not, definition does not necessarily have to have that intuitive meaning. And yes, if the condition is mathematically valuable and the kind of corresponding in some sense natural, I think natural to say that condition justification is a reasonable kind of justification. And two more kinds of justification of harm. This is a very simple kind of redundancy, what I call redundancy, just the right definition. You have a definition, D, you already accept. And then you find, OK, that something like the redundant clause, I don't need this additional condition. So obviously, because we strive for simplicity, we then use this clause, and just the right definition, because it's a condition of redundancy. And, of course, the definition and that one simplified one have to be equivalent. Again, there were many examples, and again, I think it's, as I said, reasonable. This is from the first example of the measured theory, and this is from topological resistance theory, one of the most popular, I would say. conditions of chaos, humane, chaotic, consists of three conditions, the sensitive dependence, the first, and transitivity, and having dense period points. And you find, which is quite surprising, if you access infinite, this is general, it's infinite discussion. Then transitivity and denseness period points implies already, and the sensitive dependence. So obviously many authors go on to find the notion is just in terms of transitivity And then something I think quite familiar that you say, another kind of justification

15:00 I found is that you say, the definition is just justified because it captures some free formal idea which is somehow important for describing and understanding maybe the natural word or some morals of this word. many, many examples. For instance, the notion of mixing, where you really want to capture something like when you, everyone's familiar, where you put a profit of ink in a glass of water, with time, it gets in it more, it distorts you more, and you have just a light blue mixture in the glass. And this is, for instance, what mixing tries to capture. And again, I mean, if this kind of justification, I think this kind of justification is reasonable, because We want to describe and understand that the world is important to this world. So, if you give them ideas better, then you can get this reason more. But I think it is important to see, I mean, that really not... Because these claims have obviously been made in the literature. The definitions have to be intuitive with some free form or meaning. But I think it's important, it's just... There is this kind of justification. but not at all any definition has to have with an intuitive or pre-formal meaning or culture. What do you think about the appearance of these kinds of justification information? I do think that what identified here, plus the lack of a thing named proof justification, condition justification, and the redundancy justification, the pre-formal justification, I think, are very, very widespread in methodologies. So I think if you, in the majority of these, of 10 to 10, I would say you'd find them. And, oh, sorry, the first hypothesis is not for free-normally-justified definitions, obviously, because I think for free-normally-justified definitions, just the second hypothesis holds which, if you are concerned with the letter and if it would be, like the most I was concerned with, you are also some parts of the logic of the hematocism theory. with, among other things, really describing and understanding natural world and worlds, which is what in this context I think you will find those definitions nearly always. And, I mean, of course, these are just based on my experience of Mark, my intellectual part, and my assessment of the case. So, yeah, they have all this.

17:30 And then just some quick comments about the injure relations situations. So, just very quickly, we've seen it before when you look at an argument for a definition. There can obviously be several arguments for a definition. We've already seen it before. We can see two arguments. And, you know, when you look at one argument, it's important just the same argument that you find that the kinds of justification are different. This is just important because otherwise you would always ask, why did that identify? And if they're in a way more different than a group of points in identifying what things are identified. So I think you can find definitions that are in one argument where they only prove justified or only condition justified, or only redundancy justified, or only . Like with weekly mixing, the above arguments we've seen, it's really just a number where the condition of justification . And then another point that when you look at the same definition, you look at different arguments for the same definition, you find that there can be, of course as I said before, there can be different arguments for a definition, but they can be justified in a very different way. In principle, I think every combination is just one way, like Mr. Nixon before, even all kinds of justification are identified. Like, when you look at the example of Richard, initially, as I've explained, Richard Nixon was most efficient at the beginning of the main justification, But later then, in the specials, the emergence of this kind of chaos research, when more and more the question became important of characterizing some sort of independent or randomness, then when you look at this is just a definition as before, and when you look, when you jump at the measures probability, which is very standard in some areas, some interpretations of measures of, I think, formalism. And Nixon says that almost all sufficiently past events are probabilistically independent, so expresses some kind of probabilistic independence and so justified in this way. And just, yeah, I think some more explicit criticism of that, which, of course, I think everybody says we have a justification to complement, we need them additionally to proof justification in general.

20:00 But I think there's more to say. As I said before, in order, from what Lannadoff writes, he has to be committed to claim that fields of a proof generation is the soul of a reasonable way of having the nations . And I think this seems to me wrong, because nearly every one of the people you will find that you have just all these various kinds of justification information. I think this proof justification is an agreement mechanism in the information of definitions. But it's definitely, in nearly any field, you will always find all these various kinds of justification information. And then there's the additional point, I think, here also in some sense, to take into account more the insulations between the kinds of justification and information, as I have discussed before. Because when you, at least in some parts of his book, when you read it, he very strongly asserts that the definitions are really proof-generated and suggested that they're proof-justified. It seems such as there can't really be anything else. But it's, I mean, not clear. I think it's right at that point since this part of the authority definition of measure are proof just to be proof justified. But it's not clear that it couldn't arrive at definitions in another context where it's not proof justified, or maybe a mission justified. OK, then just a quick summary of what I've done here. I've discussed how to prove in terms of the definitions, in particular given the definition, and then I've identified the other kinds of justification, which I think are performed as well, maybe this condition of justification, predominantly justification, and which have been for justification, and as I've gone, it has seemed widespread, and discussed the relations between those kinds of justification, in particular that the same definition even justified in very different ways. And the fields of interrelationship is the so important reason why exceptional and we somehow fail to say more about these interrelationships. Yeah, this is really awesome to say. Thank you and thank you for your questions.

22:30 Yeah, you said something about some mathematical trick in a proof notion that you cook up to do the trick. You said something about that, in some cases being intuitive or natural or corresponding to something natural and in other cases not and i was wondering what uh conception of natural or intuitive you're using there it's often a whole they're part of doing science to make sense of such mathematical tricks and tell a natural story behind them yeah it depends of course what you're playing for first of me what you mean by naturalness i mean i don't what i believe that And as I said, if any time can work, of course, we need to make sense of them. We have to have some kind of way of explaining what's going on and making sense of them. But what I wanted to say, it doesn't really have to be... It doesn't necessarily have to capture some kind of free-form notion. You know, you can also make sense of it by just saying, OK, I want to truth a theorem in order... This is just so important to me. I just need this kind of definition, so that's all I need. But I can make a condition just as I came. I mean, I think there are examples in the literature that the specific concept is really definition where I have doesn't have to have a very, doesn't have to have to be an intuitive meaning. And I think there's another part of the question, just to continue. Generally, I think this is the more, but in my kind of typology, I say, what is a, that's the question in general, for instance, improved justification, what is a manageable conjecture? because it's born to all develop a conjecture or condition justification, what it means. Fear of fears, especially by condition, you know, that it's equivalent in a natural way. What does this natural weight mean? And this is a very important question. And I mean, I think in my example, I think it's quite clear that it is natural. But of course, there's the wider question of somehow developing a theory about it. what does this mean to connect, and what does it mean to be a relative conjecture, does it mean a natural correspondence, and I mean, I haven't done it for a few minutes, and it would be extremely interesting, it would be extremely important for the project, and additionally, it's a different question. It did somehow answer the question. I always ask.

25:00 Yeah, I'm just curious for an example of the notion that has as yet withstood the natural story behind it, whereas it is really commonly used in mathematical scientific practice. That would be interesting. I mean, just one example I could give is when you, when you've read that, he discussed this very, very briefly in one of his last, his last example, just how they measure, and where you want to prove the extension theorem, you know, that you have some kind of, And then you make a definition of measure, in this case, whose definition makes very much sense, because it's very simple and gives the exact response, but it's not really intuitive. I mean, it's what you, the way it's correct, you're a simple two-digit. I mean, there are examples as well. So I've got a very simple example, namely, it's when you're looking at partially ordered sets, and then you decide to pay attention to those that have indescent soups for each pair, so-called order theoretic lattices. So you say, well, now we're trying as a coincidence of success to all of the planets. What sort of justification of those three is involved in making that definition? I mean, I can't say from the future .. Like in this case, I think it seems not important .. But I think you have to admit this. It wasn't .. You really have to study the literature, and especially the literature at the very beginning with the notions introduced in some way in depth to see what kind of justification was important and what kind of problems were the way it stayed in this definition was introduced. I don't know the problem context now. I think there can be many, and I would have to know the exact problem context to really see. And I don't claim that this is .. It's not all. I mean, I don't think it's . Just think, I mean, this is ..

27:30 And sometimes definitions involve generalizations of previously well-formative definitions and other times specializations. Yeah, but I think then one has to look in doubt in this generalization specializations. But yes, one has to look in, I mean, why are we interested in specializations and why are we interested in this generalization? Because, I mean, there's always a reason for wanting to be at least often the reason for wanting a generalization is not really important in specialization. For instance, you have to draw, like in some cases, you simply want that you're going to try to use an integer of just some more generalized rings. So it's a generalized definition that the results are general . And then you form it in a way that you also can prove the theory and theory, and prove it for the images, also for the . Form it in a way that you can prove it, too. It would be a condition, ah, sorry, approved by definition, which is within some kind of generalization . You know what I mean? I'm not sure that it's just . And I don't see this complete, I think just then I think it's too kind of looking what's going on in math and so important in some way that I think, although in my case, it's that I can't go into it, but in some ways you step into this situation, but you don't understand what's going on. Would you think all definitions in mathematics need justification? You said something about this, but I can imagine, in some cases, you might see what the justification is, doesn't really arrive. I mean, I think it's, of course, justification depends on words, so you have to think exactly what you think about. But I think, I mean, we want to understand what's going on in this. We have notions, for instance, which just... We don't really know why we consider them at all. And it's a possibility, and it's a situation that you may see what I would like to do. But I think there is, in most cases, there is always a very mild reason for understanding

30:00 the math. And in general, I think it's not necessarily that just from a dramatic point of view makes much sense. Because we want to be concerned with that which we understand and which gives us something that helps us. I'm not sure that this is an annual question, but only proof-justified definition. You said the definition is chosen such that the conjecture can then be proven. And then in turn, of course, the conjecture is justified because it can be proven without the definition. So isn't there some potentiality for a justification or a recircle? I mean, no, not really, because I mean, at least in this case, whatever you mean, you start with, let me say that was a conjecture. You have to conjecture the number of persons . And the justification, and at the very reasonable level, you simply find interest . But here the justification is . I think that you have to find some valuable interest that is done derived from . would you say that the pre-programming justified deposition is an explication? I mean, in general, there are so many different kinds of things. You have people with just the right definition, very right, very healthy and lacking situations, that you really think of in a successful concept. So I think it's what I mean. in the countries on the country, and if there's a place where it's extremely, let's say, you don't know how to find exactly the concept. It seems to be beautiful, so you have very, as I know, but if you can do that, necessarily,

32:30 it seems that this necessarily has to be. All right. Thank you. I think you have to start. Thank you.