In order to further prove and illustrate this point, it is necessary for us to make a comprehensive discussion on the relationship between mathematics, metaphysics and logic. In order to better understand Peirce's understanding of the relationship between metaphysics, logic and mathematics, and to better understand the difference between Peirce and the generally accepted views in this respect, we will first briefly examine Auguste Comte's division of science. In Peirce's time, Comte's scientific division was regarded as a representative standard view, if not a paradigm. Next, I will briefly introduce Peirce's own division of science and point out the difference between Peirce and Comte. After sketching out these background knowledge, I will describe in detail the mathematics, logic and metaphysics understood by Peirce, and the relationship between them. We will demonstrate why metaphysics needs logic and mathematics doesn't. In addition, we will show how mathematics and logic are different in their relationship with metaphysics.
Comte's division
In a sense, Peirce's division of science is a response to Comte's division of science in his six-volume Course of Positive Philosophy (1830- 1842). Not satisfied with the explanation of the unobservable and unverifiable reasons, Comte strictly limited empirical philosophy (or empirical science) to the description of phenomena. As he said in the tutorial, "until we finally enter an experience state, the human mind will no longer explore absolute concepts, the origin and destination of the universe, and the causes of phenomena in vain, but apply itself to the study of their laws, that is, the study of their eternal relationship with continuity and similarity." (Note: Auguste Comte's Positive Philosophy, Volume 2, translated by Harriet martino, London: john chapman Press, 1853, p. 1-2. Since then, this book has been called PP for short, with the number of volumes and the page number before and after the colon respectively. )
After such a definition of positive philosophy, Comte then divided scientism into two forms: abstract science and concrete science. The purpose of the first science is to discover the laws (or laws) existing in the phenomena we come into contact with, and the purpose of the second science is to study how these laws are applied to special situations. According to Comte's plan, abstract science includes mathematics, astronomy, physics, chemistry, biology and sociology, and each science depends on the principle of the former. Sociology focuses on the relationship between biological entities, so it depends on biological discoveries; The research object of biology is the physical object, so it depends on the discovery of physics; Physics, on the other hand, discusses objects that can be calculated, arranged and measured, so it depends on the discovery of mathematics.
Thus, in Comte's view, mathematics is an empirical science. Like physics and biology, it simply pays attention to the description of phenomena. Comte believes that geometry and mechanics should be "regarded as real natural sciences, which, like other sciences, are based on observation, although they can be systematized through extremely simplified phenomena, thus reaching a more perfect level." (PP 1:33) Comte went on to think that the phenomena studied in geometry and mechanics are "the most common, the simplest and the most abstract of all phenomena-these phenomena cannot be reduced to other phenomena, and they are completely independent of other phenomena." (PP 1:33) Therefore, geometry and mechanics should be put in the first place in scientific classification.
Comte arranged the position of mathematics in science in this way, partly because of his positivism and partly because of his view on the essence of mathematics. For Comte, mathematics is the science of measurement, or, since every measurement will be converted into numbers, mathematics is the science of numbers. In the tutorial, Comte explained that the object of mathematics is "indirect measurement of quantity, which tries to determine the quantitative relationship between them according to their exact relationship." (PP 1:38) Because mathematics is the most general theory about measurement, any research on measurable phenomena depends on mathematics. As Comte said, "any research will eventually come down to several problems." (Page 65438 +0:42)
Comte divided mathematics into concrete mathematics and abstract mathematics. For Comte, the conclusion in concrete mathematics still depends on the nature of the object under investigation, so different conclusions will be different for different objects. On the contrary, in abstract mathematics, the conclusion is completely independent of the object under investigation. For example, in concrete mathematics, two raindrops together still get a raindrop, just a bigger raindrop. On the contrary, in abstract mathematics, one plus one always equals two, whether we add two raindrops or two raincoats.
Comte's point of view is clear. For him, mathematics is an empirical or empirical science, thinking about the quantitative relationship between phenomena in the most abstract way. Assuming that empirical science regards the relationship between the continuity and similarity of phenomena as its sole goal, in Comte's view, mathematics has become "the real foundation of the whole natural philosophy" because it is just suitable for quantifying these relationships. (Page 65438 +0:32)
Peirce's division
Unlike Comte, Peirce believes that mathematics is not an empirical science. In fact, Peirce began his own scientific division by strictly distinguishing mathematics from empirical science. For Peirce, empirical science is the science of judging factual problems, that is, empirical science tries to find the results that reasoning cannot foresee alone. Pierce and Comte agree with this. But in Peirce's view, mathematics is not an empirical science because it does not pay attention to empirical facts. On the contrary, mathematics is limited to drawing inevitable conclusions from the explanations of complete assumptions, and does not care whether these explanations can be applied to practical things.
Philosophy is the most basic in empirical science, and Peirce distinguishes philosophy from specialized science. Various specialized sciences, such as quantum mechanics and molecular biology, need special background knowledge and special equipment, while philosophy studies all aspects of reality that everyone will come into contact with. According to Peirce, philosophy "is satisfied with a wider investigation and comparison of the facts of daily life. For every mature and sound person, these facts exist at every moment of his life. " (EP2: 146) (Note: Peirce Essentials, edited by Peirce, Bloomington, 1998 edition. ) Philosophy does not need special equipment, technology and background knowledge.
Peirce divided philosophy into phenomenology (which Peirce called phaneroscopy), normative science (aesthetics, ethics and logic) and metaphysics. Phenomenology studies all the phenomena that appear in our minds when we perceive, reason and dream. Standardizing science is to link these phenomena with some ideal. For aesthetics, this ideal is beauty; for ethics, this ideal is goodness; for logic, this ideal is truth. The last form of philosophy is metaphysics, which tries to explain a universal concept of the universe in order to provide some foundations for various special sciences. For Peirce, metaphysics is very close to the world view. Different from Comte, in Peirce's view, metaphysics is an empirical science. Metaphysics is followed by various specialized sciences, and Peirce is generally divided into natural science and spiritual science.
The next question is how mathematics relates to empirical science including logic and metaphysics. For many years, people have been trying to base mathematics on logic, metaphysics, phenomenology and even psychology. To understand these ideas, let's briefly examine:
1. Mathematics should be based on logic, which is the core proposition of logicism. According to its strict explanation, logicism insists that the axioms of mathematics can be derived from a set of primitive and pure logical axioms, and mathematics is actually a reasonable extension of logic. Others hold a relatively moderate position, for example, numerology is based on logic. (Note: See susan haack: "Peirce and Logicalism: A Preliminary Interpretation", published in Peirce Institution Newsletter, No.29, 1993, pp. 33-56. Susan haack thinks Peirce is a moderate logician. In addition, see Nathan Hauser: On Peirce and Logicalism-A Response to Huck, in the same period, Peirce Institution Newsletter, pp. 57-67. However, even on a very intuitive level, mathematics seems to rely on logic, because any mathematical proof seems to rely on logic to confirm its effectiveness.
Some people think that if everything is based on metaphysics, then mathematics should be based on metaphysics. Metaphysics is often regarded as the study of first principles, or according to another view, metaphysics is the most abstract study of existence. According to the first explanation, the mathematical principle is either the first principle and therefore a part of metaphysics, or the mathematical principle is a derivative principle and therefore based on metaphysics. According to the second explanation, metaphysics is regarded as the basis of mathematics because mathematical objects belong to some kind of existence.
3. Some people tend to regard phenomenology as the basis of mathematics. If phenomenology is to study phenomena that appear in our minds (whether true or false, such as dreams also belong to phenomena), then it is hard to deny that mathematical objects also belong to some kind of phenomenological objects.
Finally, some people think that because mathematics is the product of spirit, it should be based on the concrete scientific nature of psychology, while psychology studies the operation mode of spirit, its ability and limitations. However, we can also think that most people who insist on mathematical psychology do so because they (self-evident) combine one kind of psychological logic with another view that logic is the basis of mathematics. In this case, the latter view will also change into the first view.
In order to examine how Peirce views the relationship between mathematics, logic and metaphysics, I will first describe in detail what Peirce thinks mathematics, logic and metaphysics represent respectively. But because all of them are regarded as science by Peirce, it is necessary for us to discuss Peirce's understanding of science first.
For Peirce, science is not a systematic knowledge totality, nor a special method called "scientific method". In his view, science should be an attitude, that is, science is "the selfless, prudent and lifelong pursuit of knowledge;" It is a kind of persistence in truth, and the so-called truth is not the truth that human beings see now, but the truth that they can't see now. " (r 1 126: 08) (Note: All documents citing Peirce's Harvard manuscripts are subject to Robin's catalog number, followed by the page number set by the Institute of Pragmatism of Texas Tech University. The page numbers accurately reflect the order of the manuscripts in the Collected Works of Peirce, involving 33 microfilms (Cambridge, Massachusetts: Harvard University Library, 1963- 1970). See also Richard Robin: Table of Notes on Pierce (Amherst: University of Massachusetts Press, 1967 edition). As Peirce said elsewhere, science should not be "a systematic summary of a certain truth that has been proved", but "the scientific activities of science advocates" (r 17: 06). Therefore, in Peirce's view, the word "science" refers to any kind of activity, in which people who pursue science try to find the real answer to the question out of some sincere desire. In a more limited sense, science only refers to those activities that simply pursue truth. In this way, solving a murder case can be regarded as a kind of science, but in a limited sense, it does not belong to that kind of science, because its exploration of the truth is for a secondary purpose, that is, it just wants to ensure the realization of justice by finding the murderer.
mathematics
Peirce's definition of mathematics began with the definition given by his father. He defined mathematics as "the science of deducing inevitable conclusions." (Note: benjamin pierce: Linear Combinatorial Algebra, Washington D.C., 1870, section 1. ) 1895, Peirce talked about the relationship between this definition and his own definition: "The definition I advocate here is different from my father's definition. The only difference is that, in my opinion, mathematics not only includes deriving conclusions from assumptions, but also includes the framework of assumptions. " (R 18:02) Needless to say, architecture hypothesis and proof theorem are completely different things.
Peirce's expansion of mathematical definition accords with his general scientific view. In his view, science is a human activity driven by the desire to find the real answer to the question. This means that, fundamentally speaking, mathematics is what mathematicians do, and the activities of mathematicians are not limited to deducing inevitable conclusions. The inevitable conclusion of deduction is only a part of mathematics at most. Peirce keenly observed that a group of talented and influential mathematicians did a particularly poor job in this respect. Those powerful mathematical concepts were not proved for a long time, or the evidence on which these mathematical concepts were based was later proved to be false, even at that time.
In order to determine Peirce's definition of mathematics, we might as well go back a little and see how mathematics is related to empirical science. For Peirce, mathematics as a theoretical science lags behind mathematics as a practical science. In what seems to be the draft text of New Principles of Mathematics, Peirce said: "The task of mathematicians is to obtain accurate concepts and assumptions. When he first constructed these concepts and assumptions, he was inspired by a practical problem, then he described their causal relationship and finally summarized it. " (r 188: 02) In this way, when a physicist, meteorologist or economist encounters a complicated problem, he will call a mathematician for help. Peirce believes that the task of mathematicians at this time is to "imagine an ideal state of affairs different from the real state of affairs." Compared with the real state of affairs, this ideal state of affairs must be simpler, but it cannot be completely different from the real state of affairs, thus affecting the actual answer to the question. " (r 165a: 67, emphasis added) Therefore, mathematics provides a framework model for scientists, which can be considered as representing the phenomenon being studied; It is not to study the phenomenon itself with all accidental details, but only to study this model.
By defining science as the activity of practitioners, Peirce attributed the division of science to a division of labor to a great extent. Based on this attitude towards science, Peirce thinks that mathematicians are best suited to transform the loose theory of empirical facts produced by empirical research into a compact mathematical model:
The empirical results must be simplified and summarized, divorced from the facts, in order to become a perfect concept, and then can be used for mathematical purposes. In short, these empirical results must be modified to adapt to the abilities of mathematics and mathematicians. Only mathematicians know what these abilities are, so the task of constructing mathematical hypotheses must be completed by mathematicians. (r 17: 06, etc. )
So, what abilities should a qualified mathematician have? Peirce believes that a qualified mathematician should have three intellectual qualities, namely, imagination, concentration and generalization. Peirce believes that imagination is "the ability to clearly conceive complex structures for yourself"; Concentration is "the ability to grasp a problem, turn it into a form convenient for study, clarify its main points, and accurately determine what it contains and what it does not contain"; Generality refers to the ability to "realize that what seems to be a pile of complicated facts at first glance is actually a fragment of a harmonious and understandable whole" (r 252: 20) (Note: these three intellectual qualities are generally consistent with Peirce's view on concepts. As Peirce said in the article "The Law of the Mind", "Three elements constitute an idea. The first element is the inherent characteristics of ideas as an emotion. The second is its ability to influence other ideas ... The third element is the tendency of an idea to promote other ideas. " (CP Volume 6, page 135, version 1892)) In Peirce's view, generalization is "the primary ability of a mathematician" (R278A: 9 1), and it is also the most difficult skill to acquire. Peirce emphasizes imagination, concentration and generalization, which is quite different from the view that the primary task of mathematics is to provide proof.
In front of us, we described how the mathematical model appeared, and put forward some views on mathematical attitude. Next, we describe pure mathematics. Pierce defined pure mathematics as "an accurate study of ideal situations." (r 165a: 68) (Note: Peirce mentioned incidentally that this concept of mathematics is consistent with his father's definition. He said: "In 1870, Benjamin Peirce defined mathematics as' the science of deducing inevitable conclusions'. Except from perfect knowledge, we can't deduce the inevitable conclusion, and any knowledge about the real world can't be perfect. Therefore, according to this definition, mathematics must be associated with assumptions without exception. " (r 15: 1 1 and above)) That is to say, put aside the actual motivation to promote research, concentrate on studying the patterns themselves, ignore any relationship between these patterns and their external things, and study the patterns themselves completely, regardless of any motivation of the researchers. Generally speaking, pure mathematics prefers those models from which a large number of conclusions can be drawn. (r 14: 29) It is particularly important to point out that those models inspired by counting money and measuring land have produced a lot of pure mathematics.
As the discussion on the relationship between mathematics and empirical science shows, the phenomena encountered in science are an important source of mathematical concepts and theories. According to Peirce, in a more general sense, it is experience that provides mathematicians with ideas. Take mathematical concepts such as surface, line, point, straight line (Peirce is also called ray) and plane. According to Peirce:
The geometric surface is thinner than any piece of gold foil. It is as thin as the gap between a submerged stone and the water on it. The lines of geometric figures are thinner than spider silk, as narrow as a gap between a partially submerged stone and water and air. The geometric point is smaller than the needle tip, and it is as small as the gap between four perfectly matched objects. ..... From a point at the end of a line in the sky, an image is a straight line or a ray ... A plane, which remains unchanged no matter how we slide, flip or even reverse, is a geometric plane. (Rule 94, paragraph 56, of the rules of procedure)
Therefore, for Peirce, mathematical objects all come from experience and are precisely defined according to experience. But at the same time, mathematics does not care about empirical facts, which is very different from empirical science. Mathematicians regard mathematics as a fictional world, and they try to show how to use the rules in these fictional worlds to make inferences. Finally, mathematical hypothesis "is a pure spiritual creation, which only includes mathematicians' ideas or dreams, and its concern is only accuracy, clarity and consistency." "(R 17:07)
logic
For Peirce, logic is about facts, and its greatest concern is whether the premise is true or not and whether the conclusion is false. In the context of scientific research, that is, when we try to find something, it becomes a normative science. We agree with the theory that it is impossible to think that the premise is true and the conclusion is false. We oppose the theory that true premises and false conclusions can exist. Because how we conduct research is a voluntary choice, and this attitude of approval or opposition is a moral approval or opposition. (Note: Peirce: Essay, 8 volumes, edited by charles hart Shawn, Paul Weiss and Arthur buchs, Harvard University Press,1931958 edition, volume 5, page 130. This book has been referenced by CP since then. In fact, Peirce believes that reason is opposite to instinct, and it can carry out self-criticism and self-control, which is a great advantage of reason. In short, there is no instinct that we don't trust, but there is reasoning that is condemned by reasoning itself. (R 832:02) Because logic distinguishes between what it agrees with and what it condemns, it actually divides propositions into good and bad through a dichotomy. If the purpose of logic is to reproduce something, then the above distinction is transformed into the distinction between true (good reproduction) and false (bad reproduction).
Because a good method of logic research is to discover empirical truth through reasoning, logic is not limited to deductive reasoning, but also includes inductive reasoning and hypothetical reasoning. Deductive reasoning can draw definite conclusions, so it is usually regarded as the core of logic or the paradigm of reasoning, but in fact deductive reasoning can not draw any empirical knowledge alone. Therefore, it is very important to study inductive reasoning and hypothetical reasoning for logic, which is known as the foundation of metaphysics or specialized science.
In short, for Peirce, logic is a normative science. In Peirce's words, logic is "the science of how to control the principles of thought, which is for self-control and truth." (R 655: 26, 19 10) This sentence is very important. It shows that, unlike mathematics, logic is subject to reality. Because of this, logic is a normative science. In addition, as a normative science, logic is essentially different from archery (studying how to hit the target with a bow and arrow to the maximum extent). Archery studies the skills of archery, while logic studies the skills of reasoning. Of course, the learning of reasoning skills should not be confused with reasoning skills themselves. Moreover, the study of reasoning skills may not improve one's reasoning ability. In fact, Peirce deeply doubts the benefits of learning logic. In the book How to Reasoning published by 1894, Peirce said frankly: "In all sciences, the ideas produced by logic are the poorest." (R 4 13, p. 239)
The relationship between mathematics and logic
In view of the above views on mathematics and logic, the relationship between them naturally arises. At first glance, there are three possible choices: first, the laws of logic come from mathematics. Second, contrary to the first option, all mathematics comes from the laws of logic in the final analysis. Third, the relationship between mathematics and logic is just like other empirical sciences.
It is difficult to justify the view that the laws of logic come from mathematics, because mathematics and logic (at least according to Peirce's explanation) are two completely different disciplines. Logic studies how we should reason in order to make our thoughts conform to empirical truth, while mathematics is not interested in empirical truth. It is difficult to understand how mathematics becomes the basis of logic. Some people may think that logical arguments are at least a subset of mathematical arguments, that is, these arguments focus on actual facts, not just possibilities. However, this needs to show how this subset is different from the broader field of mathematics, and this can only be done by introducing the differences between logic and mathematics. Logic cannot come from any mathematical principle. In essence, mathematics can't distinguish what is experienced from what is possible. Moreover, such a path does not conform to Peirce's understanding of mathematics. In Peirce's view, mathematics is an idealization of some aspects and relationships of the phenomena we encounter. Only when mathematics is related to empirical facts can it be said that mathematics is the basis of logical empirical science. Therefore, mathematics can only be a derivative science rather than a basic science.
Can we say on the other hand that all mathematics comes from logical principles? This is the view of logicians. Peirce also opposes this second option. This view of logicism will make mathematics an empirical science, and we have no legitimate reason to limit mathematics to what can be derived from principles, which are used to ensure that our ideas represent empirical truth as much as possible. The second view is particularly problematic. Peirce repeatedly stressed that empirical truth is not the goal pursued by mathematics.
Let's look at the third point of view, which I think is also a point of view of Peirce. Here, the relationship between logic and mathematics is no different from that between other empirical sciences and mathematics. As I mentioned earlier, specialized science turned to mathematics, replacing complex problems with simpler but still representative mathematical models. It is in this way that scientific branches of mathematics have emerged, such as physics, chemistry and economics. So logicians, like physicists and economists, turn to mathematicians for help. Mathematicians absorb the materials provided by logicians and try to turn it into an ideal state of affairs, eliminate all accidental factors and replace complex relations with a simpler one. Although this simple relationship is false, it is suitable for solving the current problem. Next, mathematicians study this subsequent ideal state to see what is true. Mathematicians go further. He tried to change some characteristics of this ideal state and see where it would go. A classic example, though not logically but arithmetically, is the bold idea that the square root of negative 1 does have a definite solution, which leads to a fascinating and fruitful concept of imaginary number.