Mathematics as a science of patterns

0:00 Some of the relationships between patterns can shed light on mathematics. For brevity, I will focus on the case of pattern of curves. That is, when one pattern occurs within another. In figure 2, the sequence of stars, P, has several occurrences within pattern Q. Indeed, if we think of the sequence of stars extending infinitely, then the pattern P has infinitely many occurrences within Q. An obvious mathematical illustration of this relationship is the occurrence of the progression of even natural numbers within the entire natural number sequence. I will briefly focus on the case of the inclusion of patterns. In the second figure, the sequence of pi is repeated several times in the pattern of p. Indeed, representing the sequence of stars as a continuous infinity, when pi, the infinite number of times, is repeated in p. An obvious mathematical example is a clear natural number within a row of natural numbers. As Figure 1 shows, it's hard for me to draw examples of patterns that occur within each other, but mathematical examples are easy to find. The natural numbers occur within the non-negative rational numbers, and by coding ordered pairs of numbers as simple numbers, we can identify an occurrence of the rational within the natural. It is difficult to give examples of patterns that fit into each other, but there are actually many mathematical examples. Natural numbers fit into positive rational numbers. Copying of orderly pairs of numbers in the form of one number, we can say the connection of rational numbers to natural numbers. Pattern occurrence is a reflexive and transitive relation, which holds between structures P and Q.

2:30 When P is isomorphic to a structure R, whose positions are those of Q, and whose relations are definable through Q. Note that the last clause concerning definability is critical. This allows the structure of the natural numbers under successor to occur not only in the natural numbers under less than, where successor is a subrelation of less than, but also within structures in which a copy of the successor relation is declinable without being obtainable by merely restricting the domain of the structure. This key term allows the structure N-S not only to include N-minus, but also the subsequent member with a value less than. In addition, the subsequent member can be obtained without imposing restrictions on other structures. But the rational numbers quad, countable dense order rate, that is the rationals on less than, does not, because zero and successor are not definable in such structures. Theoretically definable from the natural numbers, less than does not occur in the natural numbers successor because the latter does not have enough positions.

5:00 There are no such terms, because zero and the following numbers are not defined by this kind of structure. On the other hand, although the real numbers can be organized with the help of the theoretical number of colors and divided by the basis of plural numbers, they do not fit with n, s. And what is s? The next one. Okay. Okay. So what is it now? In my 1981 paper and my 1997 book, I say that a pattern consists of positions that stand in various relations. This was vague, and I intentionally failed to be more precise, and still am not trying to give a mathematical account or an ontological explication of the patterns. I'm suggesting that mathematics studies patterns or does something very much like studying patterns. I'm just trying to call attention to certain features of patterns and use them to shed light on the nature of mathematics. With this in mind, I repeat, a pattern consists of positions that stand in various relations. To allow patterns to have distinguished positions, colors correspond to nomadic relations. We need something like this because national flag patterns, for example, are not just patterns consisting of various shapes. But patterns of shapes of specific colors. Orientation can also be part of the pattern. National flags are not supposed to be flown upside down. In mathematics we can have spaces with both orientations. The positive axes are directed up and to the right. And metrics, obviously more than purely logical devices, are needed to characterize patterns having these features.

7:30 This brings us to the idea of structural relativity. I am not trying to make a mathematical pattern or to explain its ontological status. Confirming that mathematics is studying a pattern or something like that, I want to draw attention to some properties of the pattern and shed light on the nature of mathematics. I repeat, the pattern consists of positions that appear in various relationships. Patterns may have only certain positions and monastic relations, for example, of color. The structure of national goods, for example, is not just a structure of real certain forms, but a form of completely different colors. Their orientation is also very important. In mathematics, it is possible to have different orientations of patterns, such as in the coordinate system, and metrics. It is obvious that one non-logical solution is not enough for the characteristics of any patterns. Hence the idea of structural relativity emerges. Section 3. Structural Relativity. The third paragraph is structural relativity. Structural relativity is simply this. The structures we can discern and describe are a function of the background devices. We have available for depicting structures. This arises whether we think of patterns and structures as kind of a mold, a format, or stencil for producing instances, or as whatever remains invariant when we apply a certain kind of transformation, or as an equivalence class or type associated with some equivalence relation.

10:00 The structures or patterns we recognize will be relative to our devices for specifying forms or transformations or equivalence relations. Furthermore, by enriching or curtailing these devices, we will obtain different notions of structure, count different things as having the same structure, and recognize different relations between structures. Geometry illustrates this nicely. It recognizes broader and broader types of structures, starting with, say, congruent figures in an oriented Euclidean plane and moving on to similar figures in a Euclidean plane without orientation, and then to figures in an affine plane or topological space. It is important when we think of patterns as a kind of stencil that allows us to form all or other specific objects, or when we talk about invariability, precision, or other transformations, or even about classes of equivalence, to be determined by the corresponding relationships. In the sense of our means or forms or relationships. Moreover, enriching or uniting the basic education, we can get different concepts of structures, different things that have the same structures and distinguish relationships between them. Geometry gives many examples here. Structural relativity can be seen when we consider the definition of pattern occurrence. It uses the notion of definability, which makes it relative to a background method of defining.

12:30 The method of defining, in turn, depends not only on the non-logical vocabulary of one's background language, but also on its logical devices. Not only do structural conditions vary with our logical background, but so do the things we take as exhibiting the same structures, and consequently, the types of structures we recognize. If we limit ourselves to describing structures as the models of various first-order schemata, then the types of structures we will define will be like the more coarse-grained ones frequently found in abstract algebra. Here one starts by defining a type of structure such as a root, a ring, or a lattice with the intention of allowing for many non-isomorphic examples of the same type. If the design structure is limited to models of various primary forms, then the types of structures will be similar to large non-verbal structures, which are often included in abstract algebra. Here you can look for determinations of such structures as a group, a ring or a grid, taking into account non-isomorphic examples of such types. As a result, most of our structural descriptions will fail to be categorical. On the other hand, using second-order schemata, we can formulate categorical descriptions of the disruptions studied by second-order number theory,

15:00 These are the terms that are considered powerful enough for most mathematical needs. Consider the additive group of integers. The integers with plus, one, and minus. We don't need to add zero as a distinguished element since we can define it as the additive identity, but we must have one and minus one as distinguished elements since they cannot be defined in terms of just addition. One example that will be more important is to take an additive group of whole numbers in the real line minus one as a separate element, because we can already determine the concept of the same thing. But you need to have both one and minus one as separate elements, because they cannot be determined by the term of the additiveness of the combination. The two-way infinite progression without distinguished elements, we could not re-identify one and minus one. To see this, consider figure three. In the sequence of stars, I have labeled A. Even being given the middle point, or zero, in the sequence B will not suffice for finding the one and the minus one, since we don't know which direction is positive.

17:30 But once we are given the positive or negative direction, we can recover the positions of 1 and minus 1 as in sequence C. Can people see that? That's what I was afraid of. Let's look at number 3, the sequence of stars marked with the letter a. If the average point is given, or 0, the sequence b is not enough to find 1 and minus 1, because we do not know which direction is positive. But if we set a positive or negative direction, we can create the position 1 and minus 1 in this sequence c. This illustrates how patterns with distinguished positions, orientations, or colors require more refined devices for specifying them. There are several patterns without distinguished positions, orientations, or codes. Section 4. Positions and their identity. You ought to parada by using even. Show figure 4, don't show figure 5 until I ask you to. Part of the trick. Key to my version of structuralism is the thesis that there is no factor to the matter of the identity of a position except with respect to the pattern to which it belongs.

20:00 Either x equals y or x is unequal to y, but otherwise there is no fact of the matter as to whether they are the same. The way in which I put this in earlier writings was something to the effect that positions are like geometrical points. They have no structure and have no identity or distinguishing features outside of the structure. The exact position of a position can only be defined by a particular pattern, the boundary of which is defined by this position. So, if we take the position x, y, the pattern P is equal to y, when x is not equal to y. I compared the position with the geometric points, with points that have no structure, that cannot be differentiated. See what I have in mind. Consider the vertices of the triangle in figure 4, not yet figure 5. Suppose we are shown figure 4 first, and it is taken for view, and then we are shown figure 5. Now, let's take figure 4 off and put on figure 5. Is the vertex C of the first triangle the same as the vertex F of the second triangle? One response might go, in the present context, that question makes no sense.

22:30 Considered as concrete drawings, we don't know whether the second triangle was drawn anew or came from the first by rotating it. In the present context, we have no way of knowing. But I want to say something stronger than this. Considered as patterns, which means, in part, disregarding the practical questions of how they are drawn, It's not that we have no way of knowing whether C is identical to F. Rather, there is nothing to know. There is no fact of the matter. If they are considered as structures independent of how and when they will be drawn, it is better to say that we simply do not have the means to find out whether C is an identical F, and this simply does not apply to D. I use this idea to solve the Nasser-esque conundrum. Mathematics fails to declare any of the definitions of numbers in terms of sets as the correct one. The Nasser Act concludes from this that numbers are not sets, indeed that they are not numbers at all. My response is that each definition picks out a different occurrence of the number pattern in a set pattern. However, there is no fact of the matter as to whether the positions in one occurrence of the number pattern I used this idea to solve a problem. Mathematics cannot count any number of numbers through the Norse curve.

25:00 It is possible to use the definition of Ternel, Von Neyman or Frege-Rassel for all other specific purposes. From this, Venocero concluded that numbers are not objects, but each definition acquires different definitions of the pattern of numbers. All of these have nothing to do with the position of the pattern of numbers and their identity in an indefinite context. Thus, the reduction of one number into a number is similar to the identification of the origin of one pattern into another. Therefore, it is wrong to think that only one form of reduction is the only truth. Moreover, it has nothing to do with whether the number is a number or not. Are numbers objects? I say they are, but not because there is a fact to their being these or those objects given to us in some independent context. Rather, I take them to be objects for the same reason Frege did, simply because they fall within the range of the first-order variables of number theory. I say yes, but not because they really are them or are objects in some kind of independent context. Rather, they can be considered objects in the same sense as the refrains, due to the fact that they fall out of the field of the first order of changes in the theory of physics. Without committing oneself to realism, one could say that the business of mathematics is to study patterns, either by discovering them or inventing them, and that talk of mathematical objects, positions, and patterns is a way of doing so. Talk of positions is a way of depicting how objects might be arranged. It is a way of converting talk of possibilities into talk of realities.

27:30 Again, I state that yes, I am a mathematician, but I do not want to defend this point of view here. In addition to the relative realism, it can be said that the task of a mathematician is to explore patterns either by opening or by construing, and the discussion of mathematical objects in the positions of patterns is not the way of such research. This talk is about how objects can be organized. A talk about possibilities can become a talk about reality. People want to hear more. It gets harder. I'm ready to go on, but I just want to know if you want to hear more. Because there are some more technical and complex things going on. Do you accept, Professor Mresnik asks, do you accept the theory that I am trying to explain? May I ask a question? I can go on. All right, I'll just go on because the translation is too hard for me to act with. An objection. An objection to the point of view that Professor Mresnik made. Some philosophers have tried to turn against me my thesis that there is no fact in the matter as to whether a position in one occurrence of a pattern is identical to a position in another occurrence of a pattern.

30:00 Their objection may be put as follows. By Leibniz's principle of the identity of indiscernible, the following should be the case. If X and Y are distinct positions of the same pattern, then there must be some relational property defiable in terms of the pattern that distinguishes them. This is true of the natural number zero, for example, which is distinguished from the others through having no predecessors. However, the additive group of integers has positions that cannot be distinguished by a relational property associated with the group. Suppose that f is a one-place predicate definable in terms of plus that does not refer to one or to minus one. Then one satisfies f if and only if minus one goes. This is not the only example. In the complex numbers i and minus i cannot be distinguished by credits definable in the structure except those that refer to i or minus i. Points in a geometrical space are similarly indistinguishable. So too are the symmetric nodes in a symmetric graph. And so are the positions in degenerate structures such as cardinalities or edgeless graphs. Which have no relations on them other than identity and non-identity.

32:30 This is not the only example. Comprehensive numbers i and minus i cannot be distinguished by the help of particles, which are divided in the structure by the light of the light, which is related to i and minus i. The points of the geometric space are also distinguished. And the symmetrical nodes of the symmetrical groups are equal to the positions of pronounced structures, such as cardinals and equidistant graphs, Citing these examples, the objection continues by claiming that I must either, one, identify plus one with minus one, plus i with minus i, and so on, or two, exclude the symmetric mathematical structures from my theory, or three, give up my thesis concerning the identity of positions in various structures. The first two alternatives are obviously unacceptable, and accepting the third would undermine my solution to Banassarath's problem. Let me divide the supposed counter-examples to my view into those concerning symmetric structures with distinguished elements. Such as plus one and minus one in the additive group of integers on one hand and those without distinguished elements such as Euclidean space on the other hand.

35:00 In putting forth examples of the first kind, my critics have ignored structural relativity. The additive group of integers is a structure with distinguished elements, and so is the complex field. Thus neither structure is a purely first or second order logical structure. We need a stronger background theory for describing such structures. Once this is appreciated, we can see that in considering isomorphisms of these structures, we must leave their distinguished elements fixed, just as we leave their other structural relations and operations fixed. Thus, this structure is not the first or second order of the structure of the logical point theory. We need a stronger finger theory to describe such a type of structure. If you understand this proposal, then it is clear that if you look at the measurements and mathematics of these structures, you should detect various elements in the same way as we detect other structural elements or operations. Interchanging plus one with minus one in the additive group does show that there is no property definable in the additive group for distinguishing plus one from minus one, except those referring to these numbers, but this is no objection to my view because the interchange also fails to be structure preserving by not preserving the distinguished elements of the structure. In short, my critics are employing a notion of structural property that is too narrow.

37:30 But this is not an objection to my point of view, since interchangeability is not a structure that is preserved by the non-preservation of undevelopable elements in this structure. In short, my critics use too narrow a concept of the functional property. Stuart Shapiro and James Ladyman have another way of showing that my critics are demanding too much. The critics demand what Klein called absolute conservability. If X and Y are distinct, they want a predicate true of one and not the other. Shapiro and Leibniz noted that Quine recognized weaker kinds of discernibility that allow us to distinguish 1 from minus 1 in relational terms. For example, 1 is greater than minus 1, but not conversely. Quine called discernibility by an asymmetric relation like this relative discernibility. But Shapiro and Nediman noticed that Quine recognized weaker types of discernibility, which allow us to distinguish 1 from minus 1, indicated above 1. For example, 1 is greater than minus 1, but on the contrary, Quine called discernibility due to asymmetrical relationship of such data relative discernibility. If they stand in a symmetric but irreflexive relation, we can weakly discern i for minus i by means of the relation of being separated by a positive distance in complex space.

40:00 Of course, if we allow numerical distinctness, that is, non-identity, to count as a relation, then any two distinct objects are weakly discernible. He also recognized the weak discernibility of the two objects if they are in symmetrical, but not reflexive, relations. We can distinguish, in a weak sense, from minus pi by the ratio of the relations that are in a positive distance from the complex of space. Of course, if it is a numerical discernibility, that is, in fact, non-tonsive. Unfortunately, these responses do not pertain to structures that have no distinct elements or those that have positions that are weakly discernible only by non-identity. The difficulty here is that my critics object to taking identity or anything definable in terms of it as a structural relation for weakly discerning. They think if two things are distinct, there must be something about them other than their being distinct that distinguishes them. My critics' thought may well be true of ordinary spatial temporal objects. But as the example of unlabeled edgeless grass shows, it's not true of every mathematical object. Whether we construe mathematical objects as positions and patterns or treat them in some other way, why should we require more of positions and patterns?

42:30 If you want to learn more about algebra, we have a lot of videos. No, on the contrary, only topologies without edges, without edges. This is completely unfair to any mathematical object, whether it is a mathematical object in the form of a position in the pattern or in other ways. Why does it require the presence of additional positions in the pattern? What we should do is forget about trying to individuate positions and cardinalities and other symmetric structures. The cardinal three has three distinct positions. This is a feature of the structure, even if it cannot be captured by some non-trivial relation among the positions. I think we shouldn't do that. We must forget about the attempt to individualize the position of cardinals and other symmetrical structures. Cardinal number 3 has three different positions. This property of the structure, even if it is not obtained, is obtained by the result of some non-trivial relations between the positions. We do not clearly recognize this property when we demand the presence of structural isomorphism not only in order to preserve the structural relationships, but in order to avoid one-minus-one, we must substitute different images with different members of this image. Conclusion. We learn from the case of symmetric structures the importance of structural relativity, as well as the introduction of distinct elements, orientations, metrics, and other devices beyond the recognizably logical ones.

45:00 These may be needed for breaking symmetries. From the cardinality examples, we learned that some properties of a structure, ones that must be preserved under isomorphism, cannot be reduced to the non-trivial properties and relations of its positions. As well as the introduction of various elements, orientation, metric and so on, in addition to the logical means of language. This may be important for the violation of symmetry. From the examples of cardinality, it follows that some properties of the structure, namely those that are preserved in isomorphism, cannot be reduced in the trivial properties or relations between the positions of classes. Thank you. As far as I understood, the notion of position is kind of basic element of what you call pattern. So, pattern is position plus certain relations they hold, right? These positions are not only questions about types of things, even if you say they are positions of other patterns, I think what any theory of patterns needs is a kind of theory of how these positions are filled or instantiated.