I will go against the trend and the “Platonic” perception of mathematics and suggest that we still consider mathematical objects from the standpoint of memetics.
I believe that the objects of mathematics did not exist until they were understood by mathematicians. Just as, say, there was no war and Peace until it was written by Tolstoy.
Mathematics is, in a broad sense, part of the cultural heritage that previous generations leave us. Mathematics is a memplex, a complex ecosystem of mutually supportive memes, the substrate for which is primarily people and secondly (we can say in a sporadic form) human means of storing information.
Another thing is that all currently known objects exist regardless of whether you personally know about them or not, since their existence is ensured, so to speak, by a peer-to-peer network of carriers.
In contrast to the platonic ideas of some absolute and static world of ideas, here it is proposed to look at the same world of ideas as something dynamic, developing and having a substrate that is vital for the survival of the idea. Somewhere it intersects with the ideas of the noosphere, but it is best invested in the general scheme of formation of cultural units.
I believe that mathematical objects can actually be destroyed, although this requires destroying all physical media that refer to these mathematical objects and somehow isolating human media so that they cannot transmit information about these objects to anyone. I would, however, call this action a suicide (by analogy with genocide) and would strongly condemn attempts to call for such a thing.
I believe that this question does not apply to mathematics in any special way, with respect to any imaginable object. Is there a vacuum? Is there love, and if so, where?
Can you hear the sound of a falling tree if there is no one around to hear it?
Let's separate digits – the symbols we use to represent numbers-and numbers-a mathematical concept. The numbers, of course, are made up by people. Modern” Arabic ” numbers were invented in India, and then borrowed by the Arabs, from where they came to Europe.
Numbers, like other mathematical concepts, are more complex. In the philosophy of mathematics, there are various concepts, ranging from radically subjective and attributing our mathematics not just to the human mind, but even more to specific versions of human culture (that is, ranging from “other mathematics in aliens” to “other mathematics in another culture”), to radically objective – from mathematics as properties of our universe (and then in another universe there could be other mathematics) to mathematics that has an independent non-physical reality to which we have access.
I personally tend to favor the latter, the neo-Platonic version, which I inexplicably like. As a fake welder, I wouldn't be able to make a very convincing case for my choice, but Roger Penrose is of the same opinion, which means I'm not in the worst company.
According to the neo-Platonic version of mathematical philosophy, mathematical concepts exist autonomously and we discover them, not invent them. When studying mathematics, it is as if we are exploring a different, ideal reality that is incomprehensibly consistent with our own (if you want, you can see evidence of a divine presence here, some see it) – as an example of cases when some mathematical objects were discovered earlier than the physical phenomena they are suitable for describing – there are several such places in string theory, although it is possible to argue – The convoluted dimensions described by Calabi-Yau spaces are rather hypothetical, but in the 60-80 years, some progress was made in describing the behavior of very real particles using string models involving long-known beta functions.
Accordingly, the simplicity of the number 3, for example, in this concept existed not only before us, as in the version of mathematics as a property of the universe, where the simplicity of the number follows from the discreteness of objects in our world and appears at the moment when, conditionally, two dinosaurs are trying to share three eggs for dinner equally, but also before the appearance of our world and
One can include the anthropic principle in the consideration of the question, among other things, and assume that intelligent life is impossible in a world whose nature does not allow us to find sufficiently consistent mathematics in it, but this is already a rather shaky assumption.
And no matter how we “scold” mathematicians and say that they come up with something that is not in the material world (well, it is not in demand by physicists, for example). anyway, mathematicians can't come up with anything that goes beyond the material world.
All the laws of mathematics that we have discovered and will discover are the laws of the material world, no matter how much we would like otherwise.
Mathematics is just a language for describing the world. All languages are man-made. It is more correct to say: there are objects in the world that are denoted using mathematics. Mathematics models some objects with the help of others. Just like an artist paints a picture of the world. The picture exists, but it reflects the world with a certain error, since it is itself a part of it.
All mathematical objects, without exception, exist before mathematicians discovered them. But it is impossible to say “where” they exist, because in the ideal world where they exist, there is no space, in our usual sense of the word. We can say that the answers to mathematical and other scientific questions are always greater than the set of questions themselves, even if we ask them endlessly.
One of the most fundamental properties of mathematics is the necessity and apodicticity of its statements. Let's start with Leibniz and the propositions of logicism developed by him (reducing the initial concepts of mathematics to the conceptual apparatus of logic).
1) The necessity of mathematical and logical truths stems from the impossibility of their logical negation, which will lead to a contradiction. To deny that “2+2=4” is to assert a contradiction.
2) Logical and mathematical truths (which are truths of reason) can be reduced in a finite number of steps to “identical truths”. For example, to prove the proposition “2+2=4” means to reduce it to the identity “1=1″. Random truths are like transfinite sets, and their reduction to” identity truths ” requires an infinite number of steps and is accessible only to God.
3) Reduction/simplication of logical and mathematical truths to “identity truths” requires the construction of a formal calculus to derive consequences from the accepted axioms.
4) Being analytical in its essence, Leibniz's criterion of necessity gives logic and mathematics an autonomous character, essentially separates them from all other sciences. The problem of substantiating logic and mathematics, unlike all other sciences, becomes a purely internal matter of these disciplines. They do not need to appeal to experience, intuition, cognitive psychology, etc.
Kant's ideas about the synthetic nature of mathematical propositions, the constructive nature of mathematical knowledge, and the independence of mathematics from logic did not shake the confidence of logicians of the late nineteenth and early twentieth centuries in the analytical nature of mathematics and logic, but they required additional clarification of the question of the relationship between logical and mathematical truths. Given the compatibility of the latter, there are three possible variants of their relationship. Either logical and mathematical truths form the same class, or logical truths are a subclass of mathematical truths, or, conversely, mathematical truths form a subclass of logical truths. According to logicists, the first and second options are incorrect, because, in their opinion, not every logical truth is mathematical. The third option remains: mathematical truths are logical truths; they are the consequences of a properly constructed logical calculus. The derivation of new mathematical truths is performed mechanically by manipulating symbols from previously selected ones, called axioms (do not forget that the true axiomatics of any consistent system contains examples that are not derived from the data in the axiom system).
Frege replaces Leibniz's reduction to an identical truth with a proof of the analytical character of the statement under consideration. To prove that mathematics is a part of logic, from the logicist point of view, means to show that all the concepts and propositions of mathematics are analytically true entities. Next, I will consider the program of justification of arithmetic (at that time it was understood as the theory of natural numbers together with the foundations of analysis) from the standpoint of logic. That is why the basic concepts of arithmetic — number, set, equality, variable, and function-have become the subject of intensive logical analysis. Let us examine the definition of “number”, which, in particular, is a” mathematical object ” and is directly related to the above question. According to Frege, the simplest and most inefficient answers to the question “what is a number?” are offered by those who believe that the meaning of the definition of a number can be established directly and does not require special methodological and, possibly, logical reflection. From this point of view, a number is either a certain psychological object, or the sign (number) that it is designated by. Psychologism and formalism in mathematics originate precisely from this vicious methodological attitude. Mathematicians say that numbers are abstracted from classes, or sets, but they do not define what exactly they mean by “abstraction “and”class”. By abstracting, we follow from objects to the concept that obey it. But a mathematician like Cantor understands abstraction as something else. For it, abstraction means creating new objects from already existing data. According to him, from the observation of five points located on the same line, we first abstract their ordering, which gives us an understanding of the meaning of the ordinal number “fifth”, and then through a new abstraction from the order in which these points are located, we get the definition of the cardinal number “five”.
However, one can easily prove the futility of any “direct” definition of a number. If a mathematician claims that a number is an idea, a sign, a whole consisting of similar parts, or the result of abstracting from many things, then one should simply ask about the applicability of such definitions in constructing mathematics as an integral science. If they cannot be applied literally or their use does not lead to a proof of the laws of arithmetic, such definitions should be considered useless. Uncertainty in the understanding of numbers creates uncertainty in the definition of other basic concepts of arithmetic. In this interpretation of the concept of a number, the equal sign cannot be used to denote an identity. Each occurrence of the number ” 5 “in the equality” 5 = 5 “will denote a different sequence of objects, and therefore the sign” = ” is not an identity sign. The name “variable” used to refer to indefinite quantities, including numbers, is itself erroneous, Frege believes. Mathematicians talk about variables as if they mean something “variable” and ” indefinite.” But this, in his opinion, is not the case. The referent of an expression can only be something constant and definite (the primary assumption in Frege's construction). Variables are fundamentally different from unit numeric terms and are used in two different ways. According to the first one, they indicate an open space where the constant can be substituted, as, for example, in the expression “x + 3”. According to the second, they function as laws, as, for example, in the equality “x+y = y + x”. In both cases, the purpose of variables is to indicate the place of occurrence of the referent, and not to indicate it. Problems with understanding variables are transferred to the concept of a function. The generally accepted definition of a function “If each value of the real variable “x”, belonging to its rank, correlates with a certain number of “y”, then y is defined as a variable and is called a function of the real variable x; y = f(x)”, according to Frege, does not stand up to criticism. Like variables, functions cannot denote indefinite numbers or quantities. Frege cites several common methodological reasons for the uncertainty and confusion in the foundations of mathematics of his time. These are tendencies to confuse a sign and what it stands for; an object and a concept; subjective (psychological) and objective (logical); to consider the meanings of signs out of their context. The latter trend is particularly widespread and dangerous. Mathematicians do not see that their science is a complex system of knowledge, in which all laws, definitions and theorems are interrelated and nothing has an independent meaning outside of this system. The meaning of mathematical terms is determined not by the representation that they evoke in our minds, but by the place that they occupy in the mathematical system; by the specific functions that they perform in it. First of all, Frege seeks to eliminate all doubts about the analytical nature of mathematical truths. If this is not done, he believes, it will be impossible to prove the objectivity and universality of mathematical laws. Although Frege did not intend, as he said, to put a new meaning in the definitions of analytical and synthetic truths, but only to interpret more precisely what other authors, and above all Kant, had in mind, the result was impressive. At the end of The Foundations of Arithmetic, Frege was even able to accuse Kant that the dichotomy of the analytic and the synthetic that he proposed was not exhaustive! The basis of Frege's classification of propositions into analytical and synthetic, a priori and a posteriori, is the position that we should only be interested in what justification should be considered the best. Such a justification, Frege believes, is deductive proof. The result of the justification depends on all the assumptions used. In a perfect justification, none of the initial premises requires proof. Premises are divided by their status into “facts” — unprovable truths about the properties of particular objects — and” universal laws ” — general statements that do not require proof in themselves. Excluding the possibility of both analytical and a posteriori propositions as contradictory by definition, Frege believes that: 1) A sentence is a priori if it is deductively deducible from a certain set of premises. Otherwise, it is a posteriori. 2) A sentence is analytical if it can be deduced from universal logical laws and definitions alone, including all statements on which their correctness depends. Otherwise, i.e., when at least one of the premises represents a judgment about a particular fact, the deduced sentence is synthetic. Thus, both analytical and synthetic propositions fall into the class of a priori truths. A posteriori sentences can only be synthetic. The revision of analytical and synthetic truths opens the way for Frege to a new purely logical understanding of the nature of mathematical truths, their universality and subordination to the laws of logic. “In fact, everything that can be an object of thought can actually be counted: the ideal and the real, concepts and things, time and space, events and events, methods and theorems; even the numbers themselves can be counted sequentially. In fact, it is not even necessary to specify the exact boundaries of the domain of the conceivable and considered, its logical completeness. From this fact, one can quickly conclude that the fundamental principles of arithmetic have nothing to do with the limited domain of objects whose distinctive features they express, just as the axioms of geometry express the distinctive features of spatial relations. On the contrary, these fundamental principles should cover everything that is “conceivable”; and a statement corresponding to this highest degree of universality should be rightly referred to the domain of logic.” According to Frege, Kant underestimated the importance of analytical truths. Like synthetic truths, they can provide new knowledge about the world. Knowledge of a few laws of arithmetic makes it possible to prove analytically the truth of arithmetic statements that are directly related to solving real practical problems. In other words, the analytical character of arithmetic truths, based on their deductive deducibility, is by no means fruitless. Together with proving the analyticity of arithmetic truths, Frege refutes their possibility of being a posteriori truths. If arithmetic truths were a posteriori, then they would be inductive truths. But the latter is impossible, because the inductive justification itself is based on the use of probability theory and thus the laws of arithmetic. “Probably, the induction procedure itself can only be justified by using general arithmetic sentences, if it is not understood as a simple habit. The latter has absolutely no power to guarantee the truth. While the scientific procedure, according to objective standards, sometimes finds a high probability justified in a single example, and sometimes considers thousands of events worthless, habit is determined by the number and strength of impressions and subjective circumstances that have no right to influence judgment. Induction must be based on the theory of probability, since it can make a proposition no more than probable. However, it is not clear how this teaching can be developed without assuming arithmetic laws.” Further, if arithmetic truths were a posteriori truths, then they would by definition depend on psychological, physiological, and physical circumstances and conditions. But in this case, mathematics would lose its universality, objectivity, and obligation. In effect, it would be self-defeating as a science, since each new empirical situation would require the creation of new laws and theorems. Finally, if mathematical truths were a posteriori, then they would be true only in the real world and would have no binding force for the possible objects of spatial intuition that belong to the domain of geometrical truths, and the necessary objects of conceivable things that belong to the domain of universal logical truths. So, Frege concludes, mathematical truths are a priori in nature. Arithmetic truths, Frege goes on to develop, cannot be synthetic truths either. In his opinion, there are only three possible sources of knowledge — observation; a priori spatial and temporal intuition; logical ability. Observation can only tell us what things really are. A priori spatial and temporal intuition tells us what things should be like if we have to imagine them in space and time. But neither observation nor intuition allows us to know what things really are when they are not observed or imagined. Knowledge of things beyond observation and imagination can only be given by our capacity for logical thinking.; Thus, all mathematical truths are a priori and analytical. Frege recognizes this conclusion as a likely but still important “correction” of Kant's point of view. Having reached this conclusion, Frege develops a critique of all definitions of the concept of number that do not satisfy the requirements of a priori and analyticity. Numbers are not properties that predicatively distinguish individual things. Attributing a number to a thing differs from specifying the color of a horse, the length of a road, or the weight of a piece of metal. To say that the leaves on this tree are green is to say something about each leaf and the foliage of the tree as a whole. To say that there are a thousand leaves on this tree means to say something that cannot be attributed either to a single leaf or to the leaves of the tree as a whole. Thus, the number is not a property of the same kind as the “green” property. Answers to the question” How much? ” require prior knowledge of what needs to be counted. When you ask a question like this, you first identify a lot of things that need to be counted — trees, cars, houses, people, money, etc. The answers to the questions “How long is this thing?”,” How much does it weigh? ” do not require such knowledge. The same set of things can be counted in different ways, and correspondingly represented by different numbers. Shoes can be counted as four shoes, as two pairs of shoes, as two right and two left shoes. It also proves that number is not an inherent property of physical things, such as extension or weight. Simple arguments against the fact that numbers are properties of things are, according to Frege, the numbers 0 and 1 — the absence of any things corresponding to the first, and the ambiguity of the second. In fact, it is impossible to definitely answer which thing corresponds to the number 0. But the question of the number 1 is no less vague, Frege argues. “We ask again: What is the point of applying the property of' one ' to any object, if, according to the understanding, each object can either be or not be one? How can a science based on such a vague concept, which has earned itself the fame of being just the most precise and precise?” But it is precisely from 1 that a natural series of numbers is generated by the successive addition of new units — the foundation of all mathematics. All things of the universe, material and ideal, real and imaginary, are counted. Therefore, numbers are universal properties. Statements about numbers are not properties of things, they are not experimental truths, they are not subjective representations, and although they function like adjectives, they are not adjectives. They exist objectively, regardless of who thinks them, outside of time and space, and are not subject to any changes. Therefore, they can only be concepts. Concepts are not subjective representations, nor are they identical to predicates. A predicate can denote a number, but only if this number falls under a certain concept. The property “The Earth has one Moon” is not a property of the idea, not of the word, but of the concept of the Moon of the Earth. Number as a concept explains why you can count physical and non-physical things separately and together; why you can form the number 0, which does not correspond to any one thing. For example, it is known that the planet Venus has no satellites. But there is a concept of the Moon of Venus, which can be attributed to the number 0 by saying “Venus has 0 moons”. Frege's conclusion is categorical: “Number is not abstracted from things by the type of color, weight, hardness, and is not a property of things in the sense that these latter are. The question still remains, to what does what is expressed by pointing to a number refer? A number is not real, but it is also not subjective, it is not a representation. A number does not arise by adding a thing to a thing. Also, nothing in this respect changes and giving the name according to each addition. The expressions 'many', 'many', and 'multiplicity' are not suitable for explaining numbers because of their vagueness.” Further, through the definition of a number as a volume of “equal-numbered concepts”, the premise of which is to equate “equal-numbered concepts” with specific logical objects. It turns out that the definition of “number” does not belong to the primary categorical dichotomy expressed in the question, which already makes an unambiguous answer to it unrealistic.
The problem of actual / potential infinity and the inclusion of these definitions in the mathematical thesaurus. Analysis of the remaining areas of the philosophy of mathematics.
Following Frege, Russell worked in this area, trying to eliminate the use of impredicative definitions with the help of “type theory”. However, his concepts of set and infinity, as well as the axiom of reducibility, turned out to be illogical. The main problem was that the qualitative differences between formal and mathematical logic were not taken into account, as well as the presence of unnecessary concepts, including intuitive ones. As a result, the theory of logicism could not eliminate the dialectical contradictions of paradoxes associated with infinity. There were only principles and methods that allowed us to get rid of at least non-predicative definitions. In his own mind, Russell was Cantor's heir.
In the late 19th and early 20th centuries, the spread of the formalist view of mathematics was associated with the development of the axiomatic method and the program of substantiation of mathematics that Hilbert put forward. The importance of this fact is indicated by the fact that the first problem of the twenty-three that he presented to the mathematical community was the problem of infinity. Formalization was necessary to prove the consistency of classical mathematics, ” while excluding all metaphysics from it.” Given the means and methods used by Hilbert, his goal was fundamentally impossible, but his program had a huge impact on all the subsequent development of the foundations of mathematics. Hilbert worked on this problem for quite a long time, initially constructing an axiomatic geometry. Since the solution of the problem turned out to be quite successful, he decided to apply the axiomatic method to the theory of natural numbers. Here is what he wrote in this regard:: “I have an important goal: it is I who would like to deal with the questions of the justification of mathematics as such, turning every mathematical statement into a strictly deducible formula.” The plan was to get rid of infinity by reducing it to a certain finite number of operations. To do this, he turned to physics with its atomism, in order to show all the inconsistency of infinite quantities. In fact, Hilbert raised the question of the relationship between theory and objective reality (a classic example of the discrepancy between the mathematical conceptual basis and the objective reality given to us in experience).
In essence, the further development of mathematics demonstrated the failure of Hilbert's program. This was already done by Godel in his early publications, who found, in fact, that dialectics is present in the process of cognition.
1) Godel showed the impossibility of a mathematical proof of the consistency of any system sufficiently extensive to include all arithmetic, a proof that would not use any other rules of inference than those available in the system itself. Such a proof, which uses a more powerful inference rule, may be useful. But if these rules of inference are stronger than the logical means of arithmetic calculus, then there will be no confidence in the consistency of the assumptions used in the proof. In any case, if the methods used are not finitist, then Hilbert's program will be impossible. Godel just shows the failure of calculations for finding a finitist proof of the consistency of arithmetic.
2) Godel pointed out the fundamental limitation of the possibilities of the axiomatic method: the Principia Mathematica system, like any other system used to construct arithmetic, is essentially incomplete, i.e. for any consistent system of arithmetic axioms, there are true arithmetic propositions that are not derived from the axioms of this system.
3) Godel's theorem shows that no extension of an arithmetic system can make it complete, and even if we fill it with an infinite set of axioms, then in the new system there will always be true propositions, but not deducible by means of this system. The axiomatic approach to the arithmetic of natural numbers is not able to cover the entire field of true arithmetic propositions, and what we understand by the process of mathematical proof is not limited to the use of the axiomatic method. After Godel's theorem, it became meaningless to expect that the concept of a convincing mathematical proof could be given once and for all outlined forms.
References:
I. N. Burova. Paradoxes of set theory and dialectics. Nauka Publ., 1976.
M.D. Potter. Set theory and its philosophy: a critical introduction. Oxford University Press, Incorporated, 2004.
Zhukov N. I. Filosofskie osnovaniya matematiki [Philosophical Foundations of Mathematics], Universitetskoe Publ., 1990.
O. M. Mizhevich. Two ways to overcome paradoxes in Cantor's set theory.
S. I. Masalova. PHILOSOPHY OF INTUITIONISTIC MATHEMATICS. Bulletin of DSTU, (4), 2006.
S. N. Tronin. A short summary of lectures on the discipline “Philosophy of Mathematics”. Kazan, 2012.
Grishin V. N., Bochvar D. A. Research on set theory and non-classical logics. Nauka Publ., 1976.
Kabakov F. A., Mendelson E. Introduction to Mathematical Logic. Nauka Publishing House, 1976.
A. V. Svetlov Philosophy of Mathematics. Basic mathematics justification programs.
G. Frege Fundamentals of arithmetic. Logical and mathematical research on the concept of number.
most mathematicians are platonists – in the sense that they believe that their objects exist in the world of ideas.(However, Plato considered only the world “real”, which few people now agree). These ideas are objective, because they are accessible and unchangeable to any mind. Thus, when studying the motion of cosmic bodies, it is impossible not to come to conic sections. On the other hand, they do not have a material carrier.�
I disagree with FARINATA UBERTI about reducibility to truths. Axiomatic theory is constructed in a different way, it �STARTS with axioms, for example, ab=ba, (a+b) c=ab+ac. a+0=a, that is, describes the properties of a given class of objects. Since these axioms do not hold, for example, for matrices, it is unnatural to call them truths.
It depends on what objects we are talking about. If you are talking about a triangle or a ball, then they exist. More precisely, there are material objects with this shape. Math just changes them a little: it makes them perfectly equal, even, and hollow. You don't always need one or the other for calculations.
And if we consider the number, then of course in nature it is not, the numbers on the road are not lying around. Therefore, counting, numbers, and functions are all human inventions. I could write a lot, but it's better if you watch a video of someone working directly in the field of mathematics: youtube.com more youtube.com and one more thing youtube.com
My opinion is that mathematical objects exist as ideas independently of mathematicians, because otherwise we would have to admit that mathematicians (and not only mathematicians, in fact, you and I too) have somehow constructed something in their brains that has no analog in the world around us. The “peculiarities of consciousness” in this case do not convince me. If ideas exist, then these very features of consciousness are just their material substrate; if they do not exist, then no features of consciousness can explain the appearance of mathematical objects in it that have no analogues in the material world.
Questions of “where” or “when” do not make sense in this case, since they appeal to the material world. There is no triangle, the number three, or the gamma function anywhere in the material world.
I will go against the trend and the “Platonic” perception of mathematics and suggest that we still consider mathematical objects from the standpoint of memetics.
I believe that the objects of mathematics did not exist until they were understood by mathematicians. Just as, say, there was no war and Peace until it was written by Tolstoy.
Mathematics is, in a broad sense, part of the cultural heritage that previous generations leave us. Mathematics is a memplex, a complex ecosystem of mutually supportive memes, the substrate for which is primarily people and secondly (we can say in a sporadic form) human means of storing information.
Another thing is that all currently known objects exist regardless of whether you personally know about them or not, since their existence is ensured, so to speak, by a peer-to-peer network of carriers.
In contrast to the platonic ideas of some absolute and static world of ideas, here it is proposed to look at the same world of ideas as something dynamic, developing and having a substrate that is vital for the survival of the idea. Somewhere it intersects with the ideas of the noosphere, but it is best invested in the general scheme of formation of cultural units.
I believe that mathematical objects can actually be destroyed, although this requires destroying all physical media that refer to these mathematical objects and somehow isolating human media so that they cannot transmit information about these objects to anyone. I would, however, call this action a suicide (by analogy with genocide) and would strongly condemn attempts to call for such a thing.
I believe that this question does not apply to mathematics in any special way, with respect to any imaginable object. Is there a vacuum? Is there love, and if so, where?
Can you hear the sound of a falling tree if there is no one around to hear it?
Let's separate digits – the symbols we use to represent numbers-and numbers-a mathematical concept. The numbers, of course, are made up by people. Modern” Arabic ” numbers were invented in India, and then borrowed by the Arabs, from where they came to Europe.
Numbers, like other mathematical concepts, are more complex. In the philosophy of mathematics, there are various concepts, ranging from radically subjective and attributing our mathematics not just to the human mind, but even more to specific versions of human culture (that is, ranging from “other mathematics in aliens” to “other mathematics in another culture”), to radically objective – from mathematics as properties of our universe (and then in another universe there could be other mathematics) to mathematics that has an independent non-physical reality to which we have access.
I personally tend to favor the latter, the neo-Platonic version, which I inexplicably like. As a fake welder, I wouldn't be able to make a very convincing case for my choice, but Roger Penrose is of the same opinion, which means I'm not in the worst company.
According to the neo-Platonic version of mathematical philosophy, mathematical concepts exist autonomously and we discover them, not invent them. When studying mathematics, it is as if we are exploring a different, ideal reality that is incomprehensibly consistent with our own (if you want, you can see evidence of a divine presence here, some see it) – as an example of cases when some mathematical objects were discovered earlier than the physical phenomena they are suitable for describing – there are several such places in string theory, although it is possible to argue – The convoluted dimensions described by Calabi-Yau spaces are rather hypothetical, but in the 60-80 years, some progress was made in describing the behavior of very real particles using string models involving long-known beta functions.
Accordingly, the simplicity of the number 3, for example, in this concept existed not only before us, as in the version of mathematics as a property of the universe, where the simplicity of the number follows from the discreteness of objects in our world and appears at the moment when, conditionally, two dinosaurs are trying to share three eggs for dinner equally, but also before the appearance of our world and
One can include the anthropic principle in the consideration of the question, among other things, and assume that intelligent life is impossible in a world whose nature does not allow us to find sufficiently consistent mathematics in it, but this is already a rather shaky assumption.
All objects of mathematics exist in the material world in the form of relations.
For example the number e determines the half life of radioactive elements
https://ru.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B8%D0%BE%D0%B4_%D0%BF%D0%BE%D0%BB%D1%83%D1%80%D0%B0%D1%81%D0%BF%D0%B0%D0%B4%D0%B0.
And no matter how we “scold” mathematicians and say that they come up with something that is not in the material world (well, it is not in demand by physicists, for example). anyway, mathematicians can't come up with anything that goes beyond the material world.
All the laws of mathematics that we have discovered and will discover are the laws of the material world, no matter how much we would like otherwise.
Mathematics is just a language for describing the world. All languages are man-made. It is more correct to say: there are objects in the world that are denoted using mathematics. Mathematics models some objects with the help of others. Just like an artist paints a picture of the world. The picture exists, but it reflects the world with a certain error, since it is itself a part of it.
All mathematical objects, without exception, exist before mathematicians discovered them. But it is impossible to say “where” they exist, because in the ideal world where they exist, there is no space, in our usual sense of the word. We can say that the answers to mathematical and other scientific questions are always greater than the set of questions themselves, even if we ask them endlessly.
One of the most fundamental properties of mathematics is the necessity and apodicticity of its statements. Let's start with Leibniz and the propositions of logicism developed by him (reducing the initial concepts of mathematics to the conceptual apparatus of logic).
1) The necessity of mathematical and logical truths stems from the impossibility of their logical negation, which will lead to a contradiction. To deny that “2+2=4” is to assert a contradiction.
2) Logical and mathematical truths (which are truths of reason) can be reduced in a finite number of steps to “identical truths”. For example, to prove the proposition “2+2=4” means to reduce it to the identity “1=1″. Random truths are like transfinite sets, and their reduction to” identity truths ” requires an infinite number of steps and is accessible only to God.
3) Reduction/simplication of logical and mathematical truths to “identity truths” requires the construction of a formal calculus to derive consequences from the accepted axioms.
4) Being analytical in its essence, Leibniz's criterion of necessity gives logic and mathematics an autonomous character, essentially separates them from all other sciences. The problem of substantiating logic and mathematics, unlike all other sciences, becomes a purely internal matter of these disciplines. They do not need to appeal to experience, intuition, cognitive psychology, etc.
Kant's ideas about the synthetic nature of mathematical propositions, the constructive nature of mathematical knowledge, and the independence of mathematics from logic did not shake the confidence of logicians of the late nineteenth and early twentieth centuries in the analytical nature of mathematics and logic, but they required additional clarification of the question of the relationship between logical and mathematical truths. Given the compatibility of the latter, there are three possible variants of their relationship. Either logical and mathematical truths form the same class, or logical truths are a subclass of mathematical truths, or, conversely, mathematical truths form a subclass of logical truths. According to logicists, the first and second options are incorrect, because, in their opinion, not every logical truth is mathematical. The third option remains: mathematical truths are logical truths; they are the consequences of a properly constructed logical calculus. The derivation of new mathematical truths is performed mechanically by manipulating symbols from previously selected ones, called axioms (do not forget that the true axiomatics of any consistent system contains examples that are not derived from the data in the axiom system).
Frege replaces Leibniz's reduction to an identical truth with a proof of the analytical character of the statement under consideration. To prove that mathematics is a part of logic, from the logicist point of view, means to show that all the concepts and propositions of mathematics are analytically true entities. Next, I will consider the program of justification of arithmetic (at that time it was understood as the theory of natural numbers together with the foundations of analysis) from the standpoint of logic. That is why the basic concepts of arithmetic — number, set, equality, variable, and function-have become the subject of intensive logical analysis. Let us examine the definition of “number”, which, in particular, is a” mathematical object ” and is directly related to the above question. According to Frege, the simplest and most inefficient answers to the question “what is a number?” are offered by those who believe that the meaning of the definition of a number can be established directly and does not require special methodological and, possibly, logical reflection. From this point of view, a number is either a certain psychological object, or the sign (number) that it is designated by. Psychologism and formalism in mathematics originate precisely from this vicious methodological attitude. Mathematicians say that numbers are abstracted from classes, or sets, but they do not define what exactly they mean by “abstraction “and”class”. By abstracting, we follow from objects to the concept that obey it. But a mathematician like Cantor understands abstraction as something else. For it, abstraction means creating new objects from already existing data. According to him, from the observation of five points located on the same line, we first abstract their ordering, which gives us an understanding of the meaning of the ordinal number “fifth”, and then through a new abstraction from the order in which these points are located, we get the definition of the cardinal number “five”.
However, one can easily prove the futility of any “direct” definition of a number. If a mathematician claims that a number is an idea, a sign, a whole consisting of similar parts, or the result of abstracting from many things, then one should simply ask about the applicability of such definitions in constructing mathematics as an integral science. If they cannot be applied literally or their use does not lead to a proof of the laws of arithmetic, such definitions should be considered useless. Uncertainty in the understanding of numbers creates uncertainty in the definition of other basic concepts of arithmetic. In this interpretation of the concept of a number, the equal sign cannot be used to denote an identity. Each occurrence of the number ” 5 “in the equality” 5 = 5 “will denote a different sequence of objects, and therefore the sign” = ” is not an identity sign. The name “variable” used to refer to indefinite quantities, including numbers, is itself erroneous, Frege believes. Mathematicians talk about variables as if they mean something “variable” and ” indefinite.” But this, in his opinion, is not the case. The referent of an expression can only be something constant and definite (the primary assumption in Frege's construction). Variables are fundamentally different from unit numeric terms and are used in two different ways. According to the first one, they indicate an open space where the constant can be substituted, as, for example, in the expression “x + 3”. According to the second, they function as laws, as, for example, in the equality “x+y = y + x”. In both cases, the purpose of variables is to indicate the place of occurrence of the referent, and not to indicate it. Problems with understanding variables are transferred to the concept of a function. The generally accepted definition of a function “If each value of the real variable “x”, belonging to its rank, correlates with a certain number of “y”, then y is defined as a variable and is called a function of the real variable x; y = f(x)”, according to Frege, does not stand up to criticism. Like variables, functions cannot denote indefinite numbers or quantities. Frege cites several common methodological reasons for the uncertainty and confusion in the foundations of mathematics of his time. These are tendencies to confuse a sign and what it stands for; an object and a concept; subjective (psychological) and objective (logical); to consider the meanings of signs out of their context. The latter trend is particularly widespread and dangerous. Mathematicians do not see that their science is a complex system of knowledge, in which all laws, definitions and theorems are interrelated and nothing has an independent meaning outside of this system. The meaning of mathematical terms is determined not by the representation that they evoke in our minds, but by the place that they occupy in the mathematical system; by the specific functions that they perform in it. First of all, Frege seeks to eliminate all doubts about the analytical nature of mathematical truths. If this is not done, he believes, it will be impossible to prove the objectivity and universality of mathematical laws. Although Frege did not intend, as he said, to put a new meaning in the definitions of analytical and synthetic truths, but only to interpret more precisely what other authors, and above all Kant, had in mind, the result was impressive. At the end of The Foundations of Arithmetic, Frege was even able to accuse Kant that the dichotomy of the analytic and the synthetic that he proposed was not exhaustive! The basis of Frege's classification of propositions into analytical and synthetic, a priori and a posteriori, is the position that we should only be interested in what justification should be considered the best. Such a justification, Frege believes, is deductive proof. The result of the justification depends on all the assumptions used. In a perfect justification, none of the initial premises requires proof. Premises are divided by their status into “facts” — unprovable truths about the properties of particular objects — and” universal laws ” — general statements that do not require proof in themselves. Excluding the possibility of both analytical and a posteriori propositions as contradictory by definition, Frege believes that: 1) A sentence is a priori if it is deductively deducible from a certain set of premises. Otherwise, it is a posteriori. 2) A sentence is analytical if it can be deduced from universal logical laws and definitions alone, including all statements on which their correctness depends. Otherwise, i.e., when at least one of the premises represents a judgment about a particular fact, the deduced sentence is synthetic. Thus, both analytical and synthetic propositions fall into the class of a priori truths. A posteriori sentences can only be synthetic. The revision of analytical and synthetic truths opens the way for Frege to a new purely logical understanding of the nature of mathematical truths, their universality and subordination to the laws of logic. “In fact, everything that can be an object of thought can actually be counted: the ideal and the real, concepts and things, time and space, events and events, methods and theorems; even the numbers themselves can be counted sequentially. In fact, it is not even necessary to specify the exact boundaries of the domain of the conceivable and considered, its logical completeness. From this fact, one can quickly conclude that the fundamental principles of arithmetic have nothing to do with the limited domain of objects whose distinctive features they express, just as the axioms of geometry express the distinctive features of spatial relations. On the contrary, these fundamental principles should cover everything that is “conceivable”; and a statement corresponding to this highest degree of universality should be rightly referred to the domain of logic.” According to Frege, Kant underestimated the importance of analytical truths. Like synthetic truths, they can provide new knowledge about the world. Knowledge of a few laws of arithmetic makes it possible to prove analytically the truth of arithmetic statements that are directly related to solving real practical problems. In other words, the analytical character of arithmetic truths, based on their deductive deducibility, is by no means fruitless. Together with proving the analyticity of arithmetic truths, Frege refutes their possibility of being a posteriori truths. If arithmetic truths were a posteriori, then they would be inductive truths. But the latter is impossible, because the inductive justification itself is based on the use of probability theory and thus the laws of arithmetic. “Probably, the induction procedure itself can only be justified by using general arithmetic sentences, if it is not understood as a simple habit. The latter has absolutely no power to guarantee the truth. While the scientific procedure, according to objective standards, sometimes finds a high probability justified in a single example, and sometimes considers thousands of events worthless, habit is determined by the number and strength of impressions and subjective circumstances that have no right to influence judgment. Induction must be based on the theory of probability, since it can make a proposition no more than probable. However, it is not clear how this teaching can be developed without assuming arithmetic laws.” Further, if arithmetic truths were a posteriori truths, then they would by definition depend on psychological, physiological, and physical circumstances and conditions. But in this case, mathematics would lose its universality, objectivity, and obligation. In effect, it would be self-defeating as a science, since each new empirical situation would require the creation of new laws and theorems. Finally, if mathematical truths were a posteriori, then they would be true only in the real world and would have no binding force for the possible objects of spatial intuition that belong to the domain of geometrical truths, and the necessary objects of conceivable things that belong to the domain of universal logical truths. So, Frege concludes, mathematical truths are a priori in nature. Arithmetic truths, Frege goes on to develop, cannot be synthetic truths either. In his opinion, there are only three possible sources of knowledge — observation; a priori spatial and temporal intuition; logical ability. Observation can only tell us what things really are. A priori spatial and temporal intuition tells us what things should be like if we have to imagine them in space and time. But neither observation nor intuition allows us to know what things really are when they are not observed or imagined. Knowledge of things beyond observation and imagination can only be given by our capacity for logical thinking.; Thus, all mathematical truths are a priori and analytical. Frege recognizes this conclusion as a likely but still important “correction” of Kant's point of view. Having reached this conclusion, Frege develops a critique of all definitions of the concept of number that do not satisfy the requirements of a priori and analyticity. Numbers are not properties that predicatively distinguish individual things. Attributing a number to a thing differs from specifying the color of a horse, the length of a road, or the weight of a piece of metal. To say that the leaves on this tree are green is to say something about each leaf and the foliage of the tree as a whole. To say that there are a thousand leaves on this tree means to say something that cannot be attributed either to a single leaf or to the leaves of the tree as a whole. Thus, the number is not a property of the same kind as the “green” property. Answers to the question” How much? ” require prior knowledge of what needs to be counted. When you ask a question like this, you first identify a lot of things that need to be counted — trees, cars, houses, people, money, etc. The answers to the questions “How long is this thing?”,” How much does it weigh? ” do not require such knowledge. The same set of things can be counted in different ways, and correspondingly represented by different numbers. Shoes can be counted as four shoes, as two pairs of shoes, as two right and two left shoes. It also proves that number is not an inherent property of physical things, such as extension or weight. Simple arguments against the fact that numbers are properties of things are, according to Frege, the numbers 0 and 1 — the absence of any things corresponding to the first, and the ambiguity of the second. In fact, it is impossible to definitely answer which thing corresponds to the number 0. But the question of the number 1 is no less vague, Frege argues. “We ask again: What is the point of applying the property of' one ' to any object, if, according to the understanding, each object can either be or not be one? How can a science based on such a vague concept, which has earned itself the fame of being just the most precise and precise?” But it is precisely from 1 that a natural series of numbers is generated by the successive addition of new units — the foundation of all mathematics. All things of the universe, material and ideal, real and imaginary, are counted. Therefore, numbers are universal properties. Statements about numbers are not properties of things, they are not experimental truths, they are not subjective representations, and although they function like adjectives, they are not adjectives. They exist objectively, regardless of who thinks them, outside of time and space, and are not subject to any changes. Therefore, they can only be concepts. Concepts are not subjective representations, nor are they identical to predicates. A predicate can denote a number, but only if this number falls under a certain concept. The property “The Earth has one Moon” is not a property of the idea, not of the word, but of the concept of the Moon of the Earth. Number as a concept explains why you can count physical and non-physical things separately and together; why you can form the number 0, which does not correspond to any one thing. For example, it is known that the planet Venus has no satellites. But there is a concept of the Moon of Venus, which can be attributed to the number 0 by saying “Venus has 0 moons”. Frege's conclusion is categorical: “Number is not abstracted from things by the type of color, weight, hardness, and is not a property of things in the sense that these latter are. The question still remains, to what does what is expressed by pointing to a number refer? A number is not real, but it is also not subjective, it is not a representation. A number does not arise by adding a thing to a thing. Also, nothing in this respect changes and giving the name according to each addition. The expressions 'many', 'many', and 'multiplicity' are not suitable for explaining numbers because of their vagueness.” Further, through the definition of a number as a volume of “equal-numbered concepts”, the premise of which is to equate “equal-numbered concepts” with specific logical objects. It turns out that the definition of “number” does not belong to the primary categorical dichotomy expressed in the question, which already makes an unambiguous answer to it unrealistic.
The problem of actual / potential infinity and the inclusion of these definitions in the mathematical thesaurus. Analysis of the remaining areas of the philosophy of mathematics.
Following Frege, Russell worked in this area, trying to eliminate the use of impredicative definitions with the help of “type theory”. However, his concepts of set and infinity, as well as the axiom of reducibility, turned out to be illogical. The main problem was that the qualitative differences between formal and mathematical logic were not taken into account, as well as the presence of unnecessary concepts, including intuitive ones. As a result, the theory of logicism could not eliminate the dialectical contradictions of paradoxes associated with infinity. There were only principles and methods that allowed us to get rid of at least non-predicative definitions. In his own mind, Russell was Cantor's heir.
In the late 19th and early 20th centuries, the spread of the formalist view of mathematics was associated with the development of the axiomatic method and the program of substantiation of mathematics that Hilbert put forward. The importance of this fact is indicated by the fact that the first problem of the twenty-three that he presented to the mathematical community was the problem of infinity. Formalization was necessary to prove the consistency of classical mathematics, ” while excluding all metaphysics from it.” Given the means and methods used by Hilbert, his goal was fundamentally impossible, but his program had a huge impact on all the subsequent development of the foundations of mathematics. Hilbert worked on this problem for quite a long time, initially constructing an axiomatic geometry. Since the solution of the problem turned out to be quite successful, he decided to apply the axiomatic method to the theory of natural numbers. Here is what he wrote in this regard:: “I have an important goal: it is I who would like to deal with the questions of the justification of mathematics as such, turning every mathematical statement into a strictly deducible formula.” The plan was to get rid of infinity by reducing it to a certain finite number of operations. To do this, he turned to physics with its atomism, in order to show all the inconsistency of infinite quantities. In fact, Hilbert raised the question of the relationship between theory and objective reality (a classic example of the discrepancy between the mathematical conceptual basis and the objective reality given to us in experience).
In essence, the further development of mathematics demonstrated the failure of Hilbert's program. This was already done by Godel in his early publications, who found, in fact, that dialectics is present in the process of cognition.
1) Godel showed the impossibility of a mathematical proof of the consistency of any system sufficiently extensive to include all arithmetic, a proof that would not use any other rules of inference than those available in the system itself. Such a proof, which uses a more powerful inference rule, may be useful. But if these rules of inference are stronger than the logical means of arithmetic calculus, then there will be no confidence in the consistency of the assumptions used in the proof. In any case, if the methods used are not finitist, then Hilbert's program will be impossible. Godel just shows the failure of calculations for finding a finitist proof of the consistency of arithmetic.
2) Godel pointed out the fundamental limitation of the possibilities of the axiomatic method: the Principia Mathematica system, like any other system used to construct arithmetic, is essentially incomplete, i.e. for any consistent system of arithmetic axioms, there are true arithmetic propositions that are not derived from the axioms of this system.
3) Godel's theorem shows that no extension of an arithmetic system can make it complete, and even if we fill it with an infinite set of axioms, then in the new system there will always be true propositions, but not deducible by means of this system. The axiomatic approach to the arithmetic of natural numbers is not able to cover the entire field of true arithmetic propositions, and what we understand by the process of mathematical proof is not limited to the use of the axiomatic method. After Godel's theorem, it became meaningless to expect that the concept of a convincing mathematical proof could be given once and for all outlined forms.
References:
I. N. Burova. Paradoxes of set theory and dialectics. Nauka Publ., 1976.
M.D. Potter. Set theory and its philosophy: a critical introduction. Oxford University Press, Incorporated, 2004.
Zhukov N. I. Filosofskie osnovaniya matematiki [Philosophical Foundations of Mathematics], Universitetskoe Publ., 1990.
O. M. Mizhevich. Two ways to overcome paradoxes in Cantor's set theory.
S. I. Masalova. PHILOSOPHY OF INTUITIONISTIC MATHEMATICS. Bulletin of DSTU, (4), 2006.
S. N. Tronin. A short summary of lectures on the discipline “Philosophy of Mathematics”. Kazan, 2012.
Grishin V. N., Bochvar D. A. Research on set theory and non-classical logics. Nauka Publ., 1976.
Kabakov F. A., Mendelson E. Introduction to Mathematical Logic. Nauka Publishing House, 1976.
A. V. Svetlov Philosophy of Mathematics. Basic mathematics justification programs.
G. Frege Fundamentals of arithmetic. Logical and mathematical research on the concept of number.
most mathematicians are platonists – in the sense that they believe that their objects exist in the world of ideas.(However, Plato considered only the world “real”, which few people now agree). These ideas are objective, because they are accessible and unchangeable to any mind. Thus, when studying the motion of cosmic bodies, it is impossible not to come to conic sections. On the other hand, they do not have a material carrier.�
I disagree with FARINATA UBERTI about reducibility to truths. Axiomatic theory is constructed in a different way, it �STARTS with axioms, for example, ab=ba, (a+b) c=ab+ac. a+0=a, that is, describes the properties of a given class of objects. Since these axioms do not hold, for example, for matrices, it is unnatural to call them truths.
It depends on what objects we are talking about. If you are talking about a triangle or a ball, then they exist. More precisely, there are material objects with this shape. Math just changes them a little: it makes them perfectly equal, even, and hollow. You don't always need one or the other for calculations.
And if we consider the number, then of course in nature it is not, the numbers on the road are not lying around. Therefore, counting, numbers, and functions are all human inventions. I could write a lot, but it's better if you watch a video of someone working directly in the field of mathematics: youtube.com more youtube.com and one more thing youtube.com
My opinion is that mathematical objects exist as ideas independently of mathematicians, because otherwise we would have to admit that mathematicians (and not only mathematicians, in fact, you and I too) have somehow constructed something in their brains that has no analog in the world around us. The “peculiarities of consciousness” in this case do not convince me. If ideas exist, then these very features of consciousness are just their material substrate; if they do not exist, then no features of consciousness can explain the appearance of mathematical objects in it that have no analogues in the material world.
Questions of “where” or “when” do not make sense in this case, since they appeal to the material world. There is no triangle, the number three, or the gamma function anywhere in the material world.