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by Corry Shores
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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]
Summary of
Charles Sanders Peirce
Collected Papers of Charles Sanders Peirce
Volume 1: Principles of Philosophy
Book 3: Phenomenology
Chapter 4: The Logic of Mathematics; An Attempt to Develop My Categories from within
§1: The Three Categories [1.417-1.421]
Brief summary:
In order to better understand mathematical necessity, we should first analyze experience in general. For, all necessary mathematical hypotheses are (somehow) obtained through experience. In particular, what we find by examining experience in general is that there are just three categories of elements of phenomena. The fact that there are no more but three will shed light on the mathematical hypothesis of number itself. The three categories of elements of phenomena are the following: {1} Qualities, like red, bitter, noble, etc. They are timeless possibilities rather than actual facts or determinate events. They are what they are without any relation to other things, but they have no perfect identities (for, to have such would require that they be understood in relation to other things). {2} Actual facts. They have temporal determination and individuality. We feel facts when we sense something outside us resisting our will. We directly sense facts, because our sensation is inherently a matter of resistance and reaction. From these experiences of secondness we infer the qualities of the external things we sense. {3} Laws or thoughts. They are not qualities, because unlike thoughts, qualities are timeless and do not admit of generation and development. Also, thoughts, unlike qualities, have reasons behind them. Thoughts or laws are not like a fact, because they are general and apply even to non-actual situations. However, thoughts and laws are general in the sense that they apply to the possible world of qualities, and they are factual in that they also apply to the actual world of actuality. The proper subject of law is thought or the mind, which is foreign to quality (of Firstness) and individual action (of Secondness), just as these two other subjects are foreign to one another.
Summary
1.417
[All necessarily hypotheses of mathematics are obtained (somehow) through experience. Therefore, in order to begin our examination of mathematical necessity, we will start by analyzing experience in general.]
[Pierce says his paper where will deal with mathematics. But, it seems from what Peirce says with regard to his distinction between logic utens and logica docens that his conern with mathematics here will be more philosophical. Here are some explanations of this terminology:
The distinction between logica utens and logica docens was drawn by medieval logicians and borrowed by Peirce. Logica utens is one’s unreflective judgment of the validity of informal arguments. It is a general idea of what good reasoning is. In contrast, logical docens is the reflective and precise rules of reasoning in formal systems” (Bunnin & Yu, 401)
According to Peirce, formal reasoning (or logica docens) is comprised of the process of deliberately making certain types of inferences at certain stages of an inquiry. On the other hand, reasoning habits (logica utens) are the acritical, and thus, non-deliberate, application of instinctive habits for making inferences. Our logica utens allows us to perform the thinking that “our regular business requires us daily to do[1].
(Chiasson)
Peirce says that specifically he will deal with the following questions about mathematics: What are the different systems of hypotheses from which mathematical deduction can set out? What are their general characters? Why are not other hypotheses possible? ... and questions like these. Normally in mathematics the problems are based on “clear and definite assumptions recognized at the outset”. These questions, however, will not have that advantage. But what makes these problems like mathematical ones is that they are concerned with possibility and necessity. And the nature of necessity is something he will examine. He then characterizes necessity in a way that reminds me of modal logic where something is necessary if it holds for all possible worlds. (See Nolt Logics section 11.2.1.) For Peirce, the necessity of mathematical hypotheses “must spring from some truth so broad as to hold not only for the universe we know but for every world that poet could create”. However, he also writes that this truth must be obtained through experience. (But I do not understand how that would happen for mathematical hypotheses, which to me would seem to be things that are not fashioned on the basis of experience but rather from pure conceptual thinking.) He then says that we should thus first examine the most universal categories and elements of all experience, “natural or poetical”. I am not exactly sure what he means by “natural or poetical,” but I assume he means either given in experience as we know it or otherwise as it could be in some possible world.]
Although the present paper deals with mathematics, yet its problems are not mere mathematical problems. It is not proposed to inquire into the methods of reasoning of mathematics particularly, although this subject will incidentally be touched upon. But mathematics performs its reasonings by a logica utens which it develops for itself, and has no need of any appeal to a logica docens; for no disputes about reasoning arise in mathematics which need to be submitted to the principles of the philosophy of thought for decision. The questions which are here to be examined are, what are the different systems of hypotheses from which mathematical deduction can set out, what are their general characters, why are not other hypotheses possible, and the like. These are not problems which, like those of mathematics, repose upon clear and definite assumptions recognized at the outset; and yet, like mathematical problems, they are questions of possibility and necessity. What the nature of this necessity can be is one of the very matters to be discovered. This much, however, is indisputable: if there are really any such necessary characteristics of mathematical hypotheses as I have just declared in advance that we shall find that there [are], this necessity must spring from some truth so broad as to hold not only for the universe we know but for every world that poet could create. And this truth like every truth must come to us by the way of experience. No apriorist ever denied that. The first matters which it is pertinent to examine are the most universal categories of elements of all experience, natural or poetical.
(227)
1.418
[There are three categories of elements of phenomena. The first are pure qualities, which can be found in every phenomenon. They have no perfect identities, and yet (somehow) each one is what it is without relation to others.]
[Peirce says that “among phenomena [there are] three categories of elements”. The first category of phenomenal elements are qualities, like “red, bitter, tedious, hard, heartrending, noble”. Peirce then notes the objection that these are not qualities of things, and thus they do not actually exist in the world but rather are merely sensations. Peirce sees this as a question of “whether we ought to say that it is the senses that make the sense-qualities or the sense-qualities to which the senses are adapted”. I am not sure if he is going to answer this or not, but he does state that “wherever there is a phenomenon there is a quality; so that it might almost seem that there is nothing else in phenomena”. He also says that qualities merge into each other; they only have likeness or partial identities and not identities of their own; some of them form systems, like colors and musical sounds; and in fact, if our experience of qualities were not so fragmentary, we might even find that there are “no abrupt demarcations between them, at all”. But despite this, “Still, each one is what it is in itself without help from the others. They are single but partial determinations”. (I have difficulty finding the overall point he is making here, or if he is simply listing features of qualities.)]
We remark among phenomena three categories of elements.
The first comprises the qualities of phenomena, such as red, bitter, tedious, hard, heartrending, noble; and there are doubtless manifold varieties utterly unknown to us. Beginners in philosophy may object that these are not qualities of things and are not in the world at all, but are mere sensations. Certainly, we only know such as the senses we effect of the evolutionary process which has made us what we are has been to blot the greater part of the senses and sensations which were once dimly felt, and to render bright, clear, and separate the rest. But whether we ought to say that it is the senses that make the sense-qualities or the sense-qualities to which the senses are adapted, need not be determined in haste. It is sufficient that wherever there is a phenomenon there is a quality; so that it might almost seem that there is nothing else in phenomena. The qualities merge into one another. They have no perfect identities, but only likenesses, or partial identities. Some of them, as the colors and the musical sounds, form well-understood systems. Probably, were our experience of them not so fragmentary, there would be no abrupt demarcations between them, at all. Still, each one is what it is in itself without help from the others. They are single but partial determinations.
(228)
1.419
[The second category of elements of phenomena are actual facts. They have temporal determination and individuality. We directly sense facts, because our sensation is inherently a matter of resistance and reaction. From these experiences of secondness we infer the qualities of the external things we sense.]
[The second category of elements of phenomena are actual facts. A quality by itself is too vague and potential to be something individual. But “an occurrence is perfectly individual”. It has a certain spatial and temporal specificity, as “It happens here and now”. (I am not sure what can be said about a quality in this regard. It would seem that a quality does not even ‘happen’. It has a timelessness of sorts to it. We discussed the temporality of firstness for example in section 1.305 and section 1.307.) Peirce then mentions “permanent facts”. I am not sure exactly what these are. Perhaps they are facts that came about and stay that way. I would not think they are like laws of physics, which I assume are thirds. He says that permanent facts “are less purely individual.” I am not sure why. Perhaps the idea is that since they are permanent, they do not have limiting temporal boundaries which would individualize them. But again, without an example it is hard for me know. He then writes that qualities are concerned with facts but they do not make up facts. So a quality might be something that occupies our minds as a result of something we perceive or interact with in the exernal world. But facts cannot be said to be made of qualities. Here we again see the non-reducibility of one structure to the others. (For this notion, see sections 1.345 to 1.347.) Facts are not simply sensed things. They are material substances in the external world that resist our will, which is why “facts are proverbially called brutal”. Qualities, however, do not resist or react, but matter does. (His next point is that we directly perceive matter. His reasoning seems to be the following. When we sense something there is a reaction and a resistance. Perhaps it is our body’s reaction to what we sense and a resistance to strong influence. But, qualities do not react or resist. So qualities are not directly involved in our sensing. For, were they involved, they would somehow participate in this reaction and resisting that is inherent to sensing. This means that we do not “infer matter from its qualities.” Instead, we directly perceive matter, and we somehow “infer qualities by generalization from what we perceive in matter”.) Peirce emphasizes that quality and fact (or action or actuality) are both elements of phenomena.]
The second category of elements of phenomena comprises the actual facts. The qualities, in so far as they are general, are somewhat vague and potential. But an occurrence is perfectly individual. It happens here and now. A permanent fact is less purely individual; yet so far as it is actual, its permanence and generality only consist in its being there at every individual instant. Qualities are concerned in facts but they do not make up facts. Facts also concern subjects which are material substances. We do not see them as we see qualities, that is, they are not in the very potentiality and essence of sense. But we feel facts resist our will. That is why facts are proverbially called brutal. Now mere qualities do not resist. It is the matter that resists. Even in actual sensation there is a reaction. Now mere qualities, unmaterialized, cannot | actually react. So that, rightly understood, it is correct to say that we immediately, that is, directly perceive matter. To say that we only infer matter from its qualities is to say that we only know the actual through the potential. It would be a little less erroneous to say that we only know the potential through the actual, and only infer qualities by generalization from what we perceive in matter. All that I here insist upon is that quality is one element of phenomena, and fact, action, actuality is another. We shall undertake the analysis of their natures below.
(228-229)
1.420
[The third category of elements are laws or thoughts. They are distinct from qualities (of Firstness) for two reasons: {1} qualities are timeless and do not admit of generation and development, while thoughts in contrast can be produced and can grow, and {2} thoughts have reasons for being what they are, but qualities do not. Laws and thoughts are distinct from facts (of Secondness) for two reasons: {a} thoughts are general in the sense that they can be given from person to person, but facts cannot be exchanged, and {b} thoughts are general in the sense that they refer to all possible things and not, like facts, just to things that happen to exist. Also, laws or thoughts cannot be reduced to a set of facts, because laws refer to more facts that can be actualized. However, even though a law is different than qualities and facts, it can be said still in a sense to be a general fact, so long as we do not call any actual fact a general fact. A law (understood as a general fact) can be said to be general, because it applies to the potential world of quality, that is to say, it applies potentially to various possible ways that the qualitative features of the world can find expression. And a law (again understood as a general fact) can be said to be factual, because it concerns the actual world of actuality. The proper subject of law is thought or the mind, which is foreign to quality (of Firstness) and individual action (of Secondness), just as these two subjects are themselves foreign to one another. ]
[The third category of elements are, from an exterior point of view, laws, but from both an interior and exterior point of view, are thoughts. (I am not sure what the conceptual significance is in this terminological distinction. I suppose were we to take an objective point of view on the workings of the world, we would call these factors that have thirdness laws, as like the laws of physics. But suppose we could somehow get inside the “mind” (perhaps in a cosmic sense) that conceives these laws or suppose that we can take the perspective of the rational structures themselves that are responsible for these laws, we would think of them as thoughts, because they are rational entities that are formulated, in some sense, conceptually. You will have to interpret that first sentence for yourself, as I am not sure I know what Peirce means by “both sides of the shield”.) (Recall again from section 1.305 and section 1.307 that) Qualities are “eternal, independent of time and of any realization”. (This means they cannot be thought of as arising and developing.) However, thoughts are things that can be “produced and grow,” and thus thoughts are not qualities. Also, thoughts have reasons for being what they are, but qualities do not. Peirce next explains why a thought is not a fact. A law or thought is general. But a fact cannot be general, in two senses. It cannot be given from one person to another. But a thought can. (I suppose the idea here is that some fact, like some collision of objects, is not something one person can give to another; for, it is an event that transpires in a certain place at a certain time, and therefore it cannot be put into another place at another time. Thoughts however can be communicated between minds. And perhaps also the idea here is that they can be instantiated in different places at different times. But that might be the next idea.) Also, a fact only refers to things that happen to exist. But a thought refers to all possible things. So for these two reasons a thought is not a fact. Peirce then notes the non-reducibility of a thought or law to a fact. (Again, for this notion, see sections 1.345 to 1.347.) He says that no collection of facts could ever constitute a law, because a law is not confined to facts but rather is determinative of how facts can be, and that determination is for all facts, even ones that would never actually occur. He then says that we can consider laws as general facts so long as we do not think that any given fact constitutes a law. For, a law has some element of potentiality to it. (Let me quote the next idea before discussing it.) “As general, the law, or general fact, concerns the potential world of quality, while as fact, it concerns the actual world of actuality”. (We just noted that a law could be seen as a general fact, so long as we do not confuse it with actual facts. Now he seems to be explaining in what sense a law can be understood as having generality and in what sense it can be seen as being factual. The part about generality is not so clear. He seems to be saying that law, being that it concerns anything possible that it is relevant to, is specifically concerned with the potential world of quality. But it is not obvious to me what the potential world of quality is, at least as it relates to laws. I would normally think that laws apply primarily to facts, and secondarily to qualities. In other words, there is some factual event in the world. Let us think of the physical event of the train whistle blowing, which is an example he has used. There is then the event (involving effort and resistance) of us perceiving that whistle. And then we experience the pure quality of the sound. But I do not understand how laws apply directly to the potential world of quality. Recall from section 1.313 that Peirce also has this idea of qualities being real things that subsist in objects, and thus for example when a woman wears a certain perfume, her own essence, like the qualities of her personality, get mixed with the qualities of the perfume: “A lady’s favorite perfume seems to me somehow to agree with that of her spiritual being. If she uses none at all her nature will lack perfume. If she wears violet she herself will have the very same delicate fineness” (157). So I am not at all certain, but to offer a guess, perhaps Peirce is saying with regard to laws and the potential world of quality that since the things in the world are expressive of certain pure qualitative essences, and furthermore, because laws govern the ways things in the world operate, that in this sense laws as general things concern the potential world of quality. That is probably not what he means, so please consult the quotation to follow. The point however, about laws as fact concerning the actual world of actuality is much easier to conceptualize, especially when we think of the example of physical laws.) Peirce’s next point seems to be that firstness, secondness, and thirdness all involve primarily different subjects. For example, action, which is the proper subject of secondness, is something foreign to mere quality, which is the subject of firstness. The peculiar subject of law is thought or mind, which is foreign to mere individual action. And given their very different subjects, quality, law, and action are all very remote from one another.]
The third category of elements of phenomena consists of what we call laws when we contemplate them from the outside only, but which when we see both sides of the shield we call thoughts. Thoughts are neither qualities nor facts. They are not qualities because they can be produced and grow, while a quality is eternal, independent of time and of any realization. Besides, thoughts may have reasons, and indeed, must have some reasons, good or bad. But to ask why a quality is as it is, why red is red and not green, would be lunacy. If red were green it would not be red; that is all. And any semblance of sanity the question may have is due to its being not exactly a question about quality, but about the relation between two qualities, though even this is absurd. A thought then is not a quality. No more is it a fact. For a thought is general. I had it. I imparted it to you. It is general on that side. It is also general in referring to all possible things, and not merely to those which happen to exist. No collection of facts can constitute a law; for the law goes beyond any accomplished facts and determines how facts that may be, but all of which never can have happened, shall be characterized. There is no objection to saying that a law is a general fact, provided it be understood that the general has an admixture of potentiality in it, so that no congeries of actions here and now can ever make a general fact. As general, the law, or general fact, concerns the potential world of quality, while as fact, it concerns the actual world of actuality. Just as action requires a peculiar kind of subject, matter, which is foreign to mere quality, so law requires a peculiar kind of subject, the thought, or, as the phrase in this connection is, the mind, as a peculiar kind of subject foreign to mere individual action. Law, then, is something as remote | from both quality and action as these are remote from one another.
(229-230)
1.421
[We can better understand the mathematical notion of number if we examine why there are only these three categories of phenomena.]
[Peirce says that we will now try to determine why there are only these three categories of phenomena. For, the reason why there are just three categories will help us understand the mathematical notion of number, which is mathematics most universal hypotheses.
Having thus by observation satisfied ourselves that there are these three categories of elements of phenomena, let us endeavor to analyze the nature of each, and try to find out why there should be these three categories and no others. This reason, when we find it, ought to be interesting to mathematicians; for it will be found to coincide with the most fundamental characteristic of the most universal of the mathematical hypotheses, I mean that of number.
(230)
From:
Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy. In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].
Or if otherwise noted:
Bunnin, Nicholas & Jiyuan Yu. The Blackwell Dictionary of Western Philosophy. Oxford: Blackwell, 2004.
Chiasson, Phyllis. “Logica Utens.” Encyclopedia Entry in the Digital Encyclopedia of Charles S. Peirce.
http://www.digitalpeirce.fee.unicamp.br
http://www.digitalpeirce.fee.unicamp.br/p-logchi.htm
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