Structuralism, symmetry & identical particles

0:00 Hawking has worked on this adverbial with his name and work in progress. So my topic is empiricism, fruit-likeness, and revision. Another tradition of logical empiricism started with the idea that we can study... Structures in science are the needs of logic, so we can use logical vocabulary, logical techniques to analyze different kinds of structures in science in relation to the language of science, structure of scientific theories, deterministic and probabilistic laws, scientific explanation, rules, interactive automation, and simplicity. These kinds of studies usually were played with questions which were not asking what kind of changes there are in these structures. So it was very useful to have an analysis of these structures and the next step was to ask what kind of changes there are in these structures. So the next period of philosophy and science was characterized by questions concerning scientific change. We had Walter Kuhn, Hanson, Ferrand and others discussing topics like theory change, scientific revolutions, what happens to meanings in research. Then we also have the emergence of what is called probability kinematics, the study of So, when the study of scientific change became popular, there was of course very much influence also from the history of science and also sociology of science, but those who were representing the traditional logical empiricism wanted also to

2:30 To apply logic in the study of scientific shapes and in the sense that when I started my career there were two important conferences in Europe which discussed these topics. One of them was organized by Polish philosophers, Terecki, Sadiapsky and Borjischki in Warsaw in 1974. The title was Formal methods in the methodology of the empirical sciences. And then in December 1977, 30 years ago, there was a conference in Helsinki, which was organized by me and my colleague Aino Huomenna, also Marki Sintonen was in the organizing committee, and the title was Logic and Epistemology of Scientific Change. The idea was to apply or to ask what kind of relevance logical techniques might have to the study of scientific change and especially to take up epistemological questions and logical science by that market. So it's interesting to start this talk by looking at what kind of research traditions were emerging. 30 years ago, as they were formulated in that conference, the proceedings of the conference have been published in the Atta-Losavina-Penny. However, there were really six groups of speakers, one drawing attention to Sunian themes, Richard Rorty, who soon became famous, was one of the speakers. Then there was the structuralist school, led by Wolfgang Stegmüller, Steve Bautzer, John Harris and Erhard Scheider, who are present in Helsinki discussing this school. Then there was the group from the London School of Economics. Representing the Popper-Lagatos approach, John Watkins, Alan Musgrave, John Worrall, and Rayan Toddy, who was then prepared with PhD at LSE on the problem of irresinulitude.

5:00 Then, Polish and Finnish logicians discussed formal questions about theory change, Serejski and Rantala among them. Tuomela and myself, we discussed the notion of truth-mindedness, and then one topic was the belief change due to new evidence of how belief systems are changed in the world of new kinds of evidence. Say that this was quite good representation of different topics at the time. I mentioned that Harry Rowland and Paul Byron were also invited to the conference but they could not attend. But really we can identify three research programs or research traditions which were included in the program. May be later represented in Germany, in terms of the regular truth-likeness program, which we actively discussed not only in Finland but in the Netherlands and in Italy especially, and then the unique revision program, very actively studied in Sweden. and also in German. So these are really European research traditions. One interesting similarity between structuralist and belief revision programs is that they both attempt to do philosophy of science and epistemology Do not apply to such entities or scientific theories as they were claimed, or they are irrelevant for the purposes of epistemology as Järvinen was. So this is something which is common to these approaches and of course the program of field-lightness has the quite opposite view that truth with other epistemic utilities like information, explanatory power, etc. is a key notion in our understanding of science and notions like knowledge and scientific knowledge.

7:30 Now, in spite of these clear differences in these programs, I think it's interesting to note that there is one important common element which has played a role in all of these programs and I would like to I would like to highlight that also in my discussion, without going too technical details, the notion of similarity has played a role in all of these programs. And by discussing these three programs, we can also see how different research conditions Interact with each other and in some cases how they pay to interact with each other. And I could add that while structuralism and truthfulness have been important topics for the philosophy of science, In formal logic, but also in artificial intelligence, so there is quite a lot of work done by computer scientists. Now, I will start by making some brief remarks on the relation of the structuralist program and the truth-like program. The classical, logical and empiricist notion of the theory plus the syntactical one, the so-called statement view, as Degler called it, the theory is a consequence class of the axion system in a form of the bandwidth disclosed under reduction.

10:00 That is the deduction. I hope there are not many missing parts in this. Within this syntax we can make a difference between observational and theoretical vocabulary. The implicit assumption with Tarski and others was that the formal language is always interpretive, so that With this syntactical entity of a theory, there is a claim which says that these statements are true in the actual world. As a structuralist, Jung instead analyzes theories as classes of structures with some intended applications. And then there are new relations between them. These applications and we can reintroduce the distinction between theoretical and observational terms and theoretical modeling models. And again there is a claim which says that all intending models belong to this class of structures. Now the semantic view of the United States. The analysis of the language-world relations, so from that perspective we have the language, we have structures, theory is a set of sentences in the language, then we have the models of the theory, and the claim is that all structures in the specific class are Models of the theory, here the theory is true in all the intended models. So the claim is that the actual world is a model of the theory of the language of the world. I'm sorry for that. So, but with this treatment we can easily reinterpret the

12:30 The theory applies to these applications, which you can say that the class of structures is the class of models of a set of assumptions of the theory, and then the claim is that all intending models belong to our models. So the problem of truth can be just very straightforward for a very pre-introduced structuralist lecture. So I don't see any real conflict with approaches. The classical work of the structuralist school is the book Archi-Techno-Technical Science in 1987 It gives interesting treatments of the reduction relation as relations between structures and then it introduces an approximation technique by using topological uniformities. So instead of having the standard class of authentic applications we have a blurred set of applications we have developed. We have a set of more flexible boundaries and then we have an approximate learning saying that these varying intently models belong to the class of the field. I think this is something which corresponds to the treatment of truth-likeness if we want to analyze the relation of theory and structure as an approximate relation. The truth-likeness problem started with the idea that it should make sense to say that the statement is close to the truth or almost true or approximately true.

15:00 There are different logical approaches with degrees of passive logic and then there is the famous attempt by Hopper in 1960 to define the notion of truth-likeness or verisimilitude in terms of Truth content and positive content are clear and this definition was reviewed by Willer and Kili in 1964 and that was the point when new proposals were in discussion and in fact the first Proposals were given in the Warsaw conference, which I mentioned, by Risto Hilkinen. He applied the notion of a sphere of similarity around possible worlds, which David Lewis had used in his treatment of counterfactual conditions. And a similarity between possible works as a primitive notion. At the same time, Dee had sent to BJP as a paper refuting Popper and proposing very simple as constituents of propositional language, propositional logic, proposing a distance between propositional constituents as an explicate of this distance between And the same idea, essentially the same idea, was then proposed by my own paper in 1975, the idea emerged to me when I listened to E. William Scott in Warsaw, and I applied it to the distance between monadic constituents in predicate logic with non-place predicates. The next steps, the hard steps, were the introduction of this measure, the distance measure, to full first-order logic, the reading with Indica's constituents, constituents in the first-order logic are trees.

17:30 The key steps in this program of truth-likeness was to replace the notion of a possible world with a linguistic description of a possible world of a constituent. They correspond to complete theories in mathematics. This is a technique. The theory as a set of possible works then corresponds to the set of constituents or the disjunction of constituents. And instead of having similarity of possible works as an undefined, primitive notion, we can introduce an explicit definition for the distance between constituents and then we can... You can study the proper measures, but to illustrate this, we call it measures. Okay, again, stimulate your imagination. Scissors are plus and minus, positive and negative. You can understand that. So, this is plus and minus, and this is the existential problem. Okay, so in a monadic language, you can have a classification system. You have a number of which are including predicates, new predicates, and then tells which new predicates are exemplified and which are non-empty and which are empty. So they look like this.

20:00 There are a number of cells and some of them are empty and some of them are non-empty. So a constituent is a statement which says for each new predicate, this is an exponential quantifier, there exists something in this cell, plus or minus. So we know that it's true for all alternatives and then the distance between two such claims. This definition can be easily defined as the number of disagreements that these constituents made about these steps. So in how many cases these disagreed, that is the distance, and I write it down after W.A. Midford in the 1970s had already used this kind of distance measure. We can then have a picture like the one here. We have here the true constituent. All other constituents are located at some distance from this center point. This is an explication of the series of neutrality that was used by Hickman. And each period is a resumption of constituent. It corresponds to some set. And then the next question is how to define the distance of this junction from the center point. The minimum distance here is of course important because the minimum is zero if this theory is true. So this minimum measures the degree of approximate truth of our theory. Pihin and Otik proposed to measure the distance of the system set by the average distance from the center point, but that has some unnatural properties.

22:30 My own first idea was to use the combination of this minimum distance and the maximum distance, but then I realized that... If the theory makes mistakes or claims, it has to pay for each mistake, so we should take the sum of all distances together and then we have the combination of what I call the mini-sum measure, which is the weighted average of the minimum and sum measures of these distances, as proposed in the DSA. So, I'm sorry, now there are telecoms, so I think we can skip that, but that was only a formalization of these different measures, minimum, maximum, average, and then the sum, and then the idea that we can combine the minimum and sum. Now, as I said, this minimum distance of a theory from a constituent is zero if and only if that theory contains that constituent. So, in the case of truth, it means that a theory has a minimum distance zero if and only if it is true. Value 0 if and only if the theory is the complete truth, so it makes no mistake, but it is identical to this centered point here. The piece of truthfulness we have among truth theories. Logical is stronger, amor, truth-like. This is what Popper required and this is what he only made. And then, among false theories, true context does not hold very logical strength.

25:00 So it doesn't reduce, maybe to binomial relations, but it matters how the false claims are located with respect to the true point. Then, because the true constituent is... If the probability distribution in a normal situation is unknown, we also need the method of estimating very similarly. My proposal has been to estimate the unknown degree of truth-likeness of a theory with respect to the true constituent C-star. If we have a probability distribution, it tells us something where the true constituent is located. We have a probability distribution over all different alternatives. Then we can calculate the expected value of that truth-likeness, in this case this sum of all truth-likeness of T, if this were the case and this is n. And this measure has, among other things, the interesting property that it estimates the tourism input of a theory to be high even, and the evidence. So the probability of the theory is zero, so it is clearly different from probability even though it uses probabilities and calculates its expected values. It works on the road to truth-likeness. I took many years to work out the basic results of this likeness to the truth, appearing in 1986, hiding the truth likeness in 1987, and then Theo Piper's edited a very nice collection of this closer to the truth, also in 1987, and I have applied these measures of truth likeness to problems in the philosophy of science.

27:30 This was a quick outline of the technical ideas and then some applications of these ideas. First we have a comparison with structuralism viewpoint from a scientific realist to the kind of Structural analysis of scientific theories by the structuralist school would be to say that if we have a theory and several intended applications instead of one big application per actual world. So if we have several applications, then it is natural to require that the theory should be truth-like relative to all intended applications. Also the comparative notion, one theory is better than another, at least in those cases when it gives a more truth-like description of all the common intended applications. So this is a kind of dominance criteria. But in many actual cases in science, my guess is that one theory would be better relative to some applications. So we could again here propose a kind of measure for the overall problem-solving capacity of a theory, its global success, and that would be the weighting sum of the degrees of truthfulness of a theory overall in the applications. And one way of using these measures. Even though the distance measures are, in my approach, defined relative constituents, so in that sense on the level of language, this distance is also in use distances between structures and in fact this way of having

30:00 Distances between full first-order logic introduces a metric for the space of models. Then we have the possibility of analyzing theory. Gierer's approach is that he says that truth is not really very interesting. Mostly because we have a theory which is an interesting question, In my approach, I started with the idea that you have a theory, you ask how similar that is to the true description of the real system. From a theory to a constituent, so I apply it this way, read here and propose it this way, and both... In both ways we have the notion of truth-likeness. A theory is truth-like if it is true in a model that is similar to the real system, or a theory is truth-like if it is similar to the most informative description of the theory.

32:30 So instead of getting rid of the notion of truth-likeness, here we introduce the idea of truth-like. Then we can also analyze idealizations, idealization of theories which contain false presuppositions, theories having, introducing or entailing counterfactuals. The method of idealization and concretization introduced by Vesek Novak and another important research program in the European philosophy of science can be analyzed in terms of these. Then we have applications to theories of reference and then to problems of approximate reasoning, explanation, reduction, prediction, analytical reasoning and more recently there has been quite a lot of abduction as an inference to the best explanation where The conclusion is not that the best explanation is true, but rather that it is truth-like on the basis of the available evidence to work on such applications of the notion of truth and its explanation as we have learned by the hour. For comparative purposes, we can say that the step from one theory to another is progressive and progressive relative to the problem if the new theory is more truth-like.

35:00 It makes sense with applications because this notion of estimating very similar to another seems progressive on applications of actuality. Theories from the history of science and the evidence needed may include the present. So this is the treatment of theory change. This is shown on a mini-sum measure. Examples of progressive theory change include cases where we learn new truths, we have simulated truths, we have old truth theories. We may have cases where from ignorance to Something which is false but sufficiently close to the truth, that is also a kind of progress. Then we have the properest truth content principle, that if we take the truth content of a theory, it is better than the original theory, if the theory is false. The language is changing so that we can introduce deeper and deeper constituents and theories by findings of conceptual enrichment. So these are examples of theory, revision, theory change in terms of all these two types of questions.

37:30 Now I come to a comparison with belief revision. However, formal study was influenced by the theory of counterfactuals because there, if it were again, this kind of counterfactual is acceptable by Ramsey's test if and only if the minimum change of our belief system needed to accept that season A also requires changes of applications of the theory changing side. The theory of codes, new legislation, and then later in databases of computer science, the theory of, exposed by, with the other force, again I have to say, by the 80s, behind syntactical theory, the belief system is closed under, and this is motivated by Isaac Levi, the accepted information model, accepts the game that is committed, the contents of, and then the problem of,

40:00 The first question is how to maintain consistency by minimal. We have new evidence to preserve as many quadratic formulations in cases that are not closed. The 8MG approach has three main theories. We have a belief system and we add this and take the reductive law. Then we have the vision in cases where they may contradict K and there we have reduced the idea of a minimal change. And then we have contraction by A, which means that we retract from the A the sentence A, but that is not enough, we have to retract also all sentences with A in them. The so-called B-by entity says that revision can be defined in terms of contraction and expansion, so revision by A means that we first make room for A by These are the basic rules and now we come to the treatment of similarities. As we know, similarity has been an important part of structural reform in terms of pathological

42:30 In the belief revision, a treatment of this minimality of the change of the belief system was then formalized by Adam Rowe in 1988. We apply the rules technique of spheres of similarity and in that case we center these similarity spheres around the original belief system K. If we define for K that S is the smallest sphere that intersects A.