Hipparchia
Primus registratum
On Simplicity
by Hipparchia
From my study of Newton I come away unclear and perplexed about many things. I would like to ask many questions, and learn how to formulate many others. My general sense of his work so far is at best fuzzy and incomplete. Many propositions from the first and third books were brilliant, handsome and fun to work out; others unclear, confusing or even unintelligible, at times. Working out an alternative proof to some proposition would not be an option to me, because I do not feel I have reached a sufficient amount of understanding so as to go beyond any given proof(s) to a proposition and conceive of alternative paths. If I could work out something like that, at some point, it would be thrilling. But lacking in understanding and feeling generally more inclined (or limited) toward a somewhat more abstract and philosophical side of mathematics (and mathematical physics), I feel that I should limit myself to what I can formulate meaningful questions about. Such, I think, are the first couple of rules we study in the Principia.
I would like to talk about the first two rules because they seem to me to be correct yet in some way questionable. This sounds contradictory at first, which is why I feel like exploring this dichotomy in my thoughts about these rules. I feel that they are questionable not in the sense that I see alternative answers (or rules) in my mind, but questionable in the sense of: why do we hold these rules to be guiding principles in our study of nature? What is the appeal? In other words, I wish to explore how we come to recognize principles such as Newton’s first two rules rather than question their validity. I emphasize “recognize” because it seems to me to be what happens when we see that he is correct in pointing out what should be our guiding principles of investigation.
Newton states in his first rule: “there ought not to be admitted any more causes of natural things than those which are both true and sufficient to explain their phenomena.” In the second one he advises: “Accordingly, to natural effects of the same kind the same causes should be assigned, as far as possible.” There is something fundamentally correct and powerful about these statements that goes beyond my ability to give a logical account. On one hand, I feel the rules are self-explanatory because the opposite to both strikes me as redundant and even absurd: more causes for natural things than is necessary and sufficient. Or an indefinite number of causes for a certain limited number of effects, and vice-versa. One could say there is something more powerful about what is not redundant (grammatical examples like “this is a flower, and it is red” and “this is a red flower” come to mind). And the rules warn us of the dangers of redundancy, which cold perhaps include: loss of meaning, power, truth, and simplicity.
The way these rules are related is that they both say something about the way we should go about doing science. To quote the philosophers’ position according to Newton: “Nature is simple, and does not indulge herself in superfluous causes.” We look at our own beings and often find that we do not understand the moral, intellectual, emotional and sexual aspects of ourselves. Yet we are also tempted to think of nature as being in a way simple. Perhaps there is a tendency in us that makes it difficult to understand that simplicity in nature, even though we may have a strong sense of it. Language, religion, superstition, culture, society, institutions and various other kinds of other constructs perhaps hinder our ability to see the simplicity in nature, and make us instead come up with methods and languages foreign to nature that may divert us from a truer understanding of it. Considering the options available, one would correctly choose the path of science and philosophy. But might there be a danger even to this? Could one reach a point where one can no longer tell where science ends and where wishful thinking and metaphysics start? Where to draw the line between true science and metaphysics?
Newton does not want us to admit causes that are not “both true and sufficient”. One way to take this is as a suggestion on how to succeed in playing the game of science well, in the sense that we do not want to complicate things by saying that a more complex and technical explanation is just as good as a simpler one, when it could be that one explanation would be held at the sacrifice of the effectiveness and beauty characteristic of the other? The rules can also be seen as reminders that, in practicing science, not multiplying causes unnecessarily is conducive to clarity, consistency and correctness. But the question still remains for me whether we can know to what degree nature is infallibly orderly (and in this sense, simple), clear and consistent, and to what degree we are the originators of this regularity. Experience never indeed indicates to us anything less than infallible consistency in its workings. And in so far as we experience nature as infallible, one could say that nature is in that respect simple. It may be relevant here to remind ourselves of one of Einstein’s popular quotes: God does not play dice. In other words, there is an infallible order to things.
We see first with Ptolemy (I believe) how he makes use of the principle of simplicity. He observes: “In general, we consider it a good principle to explain the phenomena by the simplest hypotheses possible, in so far as there is nothing in the observations to provide a significant objection to such a procedure” (Toomer, 136). It would seem that simplicity is a helpful tool in formulating theories and testing propositions about nature. But what do we mean by “simple” or “simplicity” when we talk about how to apply it as a tool to our study of nature? Isolated, the terms sound general and vague, and examples and analogies could perhaps be the only route to an understanding of what we mean by “simple”. If we contrast Euclidian and Newtonian propositions (from the first book only), the distinction is primarily in the way the propositions are presented. In Euclid they are always step-wise, strictly geometrical and clearly laid out; in Newton, not everything is presented to the reader, certain empirically derived concepts (like that of force) are tied to geometrical ones in ways that are not so obvious and indisputable. I want to go so far as to claim that there is a certain beauty to Euclid’s propositions that is absent in Newton, which could be due to the necessary complexity of Newton’s work. Their aims are different, and Newton is building on Euclid (and Apollonius) and going beyond him as he attempts to relate the a posteriori truths of nature to the a priori truths of geometry, using both observation and pure geometrical relations to bring them to coincide.
(Attempting to answer the question of what we mean by simplicity seems to me to be analogous to: what do you mean by knowledge, virtue, and justice? {in the sense of what are...?} The reason for this analogy is because these ‘objects’ of investigation appear to be of a similar kind: knowledge, virtue, simplicity, beauty, time, etc. Thus I am skeptical as to how far we can go if we approach them in the form of: what is...what does it mean...How do you define...., etc.1)
Euclid helps us see that we have a preference for clarity and simplicity in our communication of ideas. Achieving clarity of thought is another way to say that one understands things in a simplified way. (And I think it this achievement that seems to be what makes life meaningful and worthwhile.)2 We could almost see our preference for order and simplicity as a craving of some sort—a craving to understand, most likely. This natural tendency that we experience in our daily lives can be seen from the way we organize our wardrobes, our books, our own rooms, and our thoughts to the way we look at the world beyond our sense perception. This need to order things by shelving them, like books, in a way that is workable, convenient, simple for us to use, may be at the core of what drives us to do anything at all.3 It may be our way of being. A way of being that is tightly related to the search for knowledge, for truth.4 There is a natural tendency in humans to believe more easily what is presented to us as simple, clear and orderly rather than what appears to be complex, obscure, or chaotic. Perhaps we are “wired” in such a way that everything we clearly understand (or feel) is simple to us in the sense that we know it.5
If we imagine some sort of method or device which we could call a scale of simplicity (i.e. one that would allow us to measure the “unit” of simplicity for every proposition/truth-claim—assuming such thing as a “unit of simplicity” to be a measurable quantity), one could then imagine that at some point down the scale of simplicity, the simple becomes almost analogous to the true, or the self-evident. The further down we go in enumerating simple things on this range of simplicity, the easier it becomes to identify statements as true. Experience again helps us see here how simpler propositions (truth-claims) are easier to see and agree on than more complex ones. Thus it(experience) teaches us to see that a proposition is easier understood to the extent that it is simple. I am not sure how intelligible this thought experiment could be to others, but I feel that insight into a true claim makes that claim(whatever it may be) a simple one. I am trying to talk about an insight myself here, and I am not sure I have succeeded, but I feel that what I want to say is this: the simple is analogous to the true.
1. I do not wish to go any further with this and make references to authors who not all of us study here, such as Wittgenstein, for instance, who has an attractive approach to these kind of questions, which for him, very roughly, are not legitimate questions, but ‘questions’ that arise from a certain misunderstanding of the roughly, are not legitimate questions, but ‘questions’ that arise from a certain misunderstanding of the distinction between words of different uses. He calls words such as time, knowledge, meaning, etc. “odd-job” words to which we mistakenly apply the same form of questioning as we do to “what is a dog?” to which the answer is unequivocally: This!” (by pointing out to a dog). If I’m wrong about this, however, help me see it, because I’m not sure so far that talking about “the simple” is any different from talking about “the virtuous”.
2. Is this unintelligible, a stretch, or do you see what I mean?
3.A propos, I was struck by your house in this respect earlier today; how natural and comfortable it felt moving around such a pleasant little house where everything so simple and unassuming was where it belonged—I don’t know how I could answer: how do you know/learn where things belong?--as opposed to the uncomfortable feeling I remember getting a year and a half earlier in your trailer, trying to use the restroom and even just moving around. Not only how we arrange our furniture, but how we even built our houses is indicative of a tendency to order and simplicity.
4. This may look like an unaccounted for assumption, but I am thinking of it as something obvious, something that is proved daily, and not just in the laboratory...
5. Does this make sense? I’m not sure how to explain it
by Hipparchia
From my study of Newton I come away unclear and perplexed about many things. I would like to ask many questions, and learn how to formulate many others. My general sense of his work so far is at best fuzzy and incomplete. Many propositions from the first and third books were brilliant, handsome and fun to work out; others unclear, confusing or even unintelligible, at times. Working out an alternative proof to some proposition would not be an option to me, because I do not feel I have reached a sufficient amount of understanding so as to go beyond any given proof(s) to a proposition and conceive of alternative paths. If I could work out something like that, at some point, it would be thrilling. But lacking in understanding and feeling generally more inclined (or limited) toward a somewhat more abstract and philosophical side of mathematics (and mathematical physics), I feel that I should limit myself to what I can formulate meaningful questions about. Such, I think, are the first couple of rules we study in the Principia.
I would like to talk about the first two rules because they seem to me to be correct yet in some way questionable. This sounds contradictory at first, which is why I feel like exploring this dichotomy in my thoughts about these rules. I feel that they are questionable not in the sense that I see alternative answers (or rules) in my mind, but questionable in the sense of: why do we hold these rules to be guiding principles in our study of nature? What is the appeal? In other words, I wish to explore how we come to recognize principles such as Newton’s first two rules rather than question their validity. I emphasize “recognize” because it seems to me to be what happens when we see that he is correct in pointing out what should be our guiding principles of investigation.
Newton states in his first rule: “there ought not to be admitted any more causes of natural things than those which are both true and sufficient to explain their phenomena.” In the second one he advises: “Accordingly, to natural effects of the same kind the same causes should be assigned, as far as possible.” There is something fundamentally correct and powerful about these statements that goes beyond my ability to give a logical account. On one hand, I feel the rules are self-explanatory because the opposite to both strikes me as redundant and even absurd: more causes for natural things than is necessary and sufficient. Or an indefinite number of causes for a certain limited number of effects, and vice-versa. One could say there is something more powerful about what is not redundant (grammatical examples like “this is a flower, and it is red” and “this is a red flower” come to mind). And the rules warn us of the dangers of redundancy, which cold perhaps include: loss of meaning, power, truth, and simplicity.
The way these rules are related is that they both say something about the way we should go about doing science. To quote the philosophers’ position according to Newton: “Nature is simple, and does not indulge herself in superfluous causes.” We look at our own beings and often find that we do not understand the moral, intellectual, emotional and sexual aspects of ourselves. Yet we are also tempted to think of nature as being in a way simple. Perhaps there is a tendency in us that makes it difficult to understand that simplicity in nature, even though we may have a strong sense of it. Language, religion, superstition, culture, society, institutions and various other kinds of other constructs perhaps hinder our ability to see the simplicity in nature, and make us instead come up with methods and languages foreign to nature that may divert us from a truer understanding of it. Considering the options available, one would correctly choose the path of science and philosophy. But might there be a danger even to this? Could one reach a point where one can no longer tell where science ends and where wishful thinking and metaphysics start? Where to draw the line between true science and metaphysics?
Newton does not want us to admit causes that are not “both true and sufficient”. One way to take this is as a suggestion on how to succeed in playing the game of science well, in the sense that we do not want to complicate things by saying that a more complex and technical explanation is just as good as a simpler one, when it could be that one explanation would be held at the sacrifice of the effectiveness and beauty characteristic of the other? The rules can also be seen as reminders that, in practicing science, not multiplying causes unnecessarily is conducive to clarity, consistency and correctness. But the question still remains for me whether we can know to what degree nature is infallibly orderly (and in this sense, simple), clear and consistent, and to what degree we are the originators of this regularity. Experience never indeed indicates to us anything less than infallible consistency in its workings. And in so far as we experience nature as infallible, one could say that nature is in that respect simple. It may be relevant here to remind ourselves of one of Einstein’s popular quotes: God does not play dice. In other words, there is an infallible order to things.
We see first with Ptolemy (I believe) how he makes use of the principle of simplicity. He observes: “In general, we consider it a good principle to explain the phenomena by the simplest hypotheses possible, in so far as there is nothing in the observations to provide a significant objection to such a procedure” (Toomer, 136). It would seem that simplicity is a helpful tool in formulating theories and testing propositions about nature. But what do we mean by “simple” or “simplicity” when we talk about how to apply it as a tool to our study of nature? Isolated, the terms sound general and vague, and examples and analogies could perhaps be the only route to an understanding of what we mean by “simple”. If we contrast Euclidian and Newtonian propositions (from the first book only), the distinction is primarily in the way the propositions are presented. In Euclid they are always step-wise, strictly geometrical and clearly laid out; in Newton, not everything is presented to the reader, certain empirically derived concepts (like that of force) are tied to geometrical ones in ways that are not so obvious and indisputable. I want to go so far as to claim that there is a certain beauty to Euclid’s propositions that is absent in Newton, which could be due to the necessary complexity of Newton’s work. Their aims are different, and Newton is building on Euclid (and Apollonius) and going beyond him as he attempts to relate the a posteriori truths of nature to the a priori truths of geometry, using both observation and pure geometrical relations to bring them to coincide.
(Attempting to answer the question of what we mean by simplicity seems to me to be analogous to: what do you mean by knowledge, virtue, and justice? {in the sense of what are...?} The reason for this analogy is because these ‘objects’ of investigation appear to be of a similar kind: knowledge, virtue, simplicity, beauty, time, etc. Thus I am skeptical as to how far we can go if we approach them in the form of: what is...what does it mean...How do you define...., etc.1)
Euclid helps us see that we have a preference for clarity and simplicity in our communication of ideas. Achieving clarity of thought is another way to say that one understands things in a simplified way. (And I think it this achievement that seems to be what makes life meaningful and worthwhile.)2 We could almost see our preference for order and simplicity as a craving of some sort—a craving to understand, most likely. This natural tendency that we experience in our daily lives can be seen from the way we organize our wardrobes, our books, our own rooms, and our thoughts to the way we look at the world beyond our sense perception. This need to order things by shelving them, like books, in a way that is workable, convenient, simple for us to use, may be at the core of what drives us to do anything at all.3 It may be our way of being. A way of being that is tightly related to the search for knowledge, for truth.4 There is a natural tendency in humans to believe more easily what is presented to us as simple, clear and orderly rather than what appears to be complex, obscure, or chaotic. Perhaps we are “wired” in such a way that everything we clearly understand (or feel) is simple to us in the sense that we know it.5
If we imagine some sort of method or device which we could call a scale of simplicity (i.e. one that would allow us to measure the “unit” of simplicity for every proposition/truth-claim—assuming such thing as a “unit of simplicity” to be a measurable quantity), one could then imagine that at some point down the scale of simplicity, the simple becomes almost analogous to the true, or the self-evident. The further down we go in enumerating simple things on this range of simplicity, the easier it becomes to identify statements as true. Experience again helps us see here how simpler propositions (truth-claims) are easier to see and agree on than more complex ones. Thus it(experience) teaches us to see that a proposition is easier understood to the extent that it is simple. I am not sure how intelligible this thought experiment could be to others, but I feel that insight into a true claim makes that claim(whatever it may be) a simple one. I am trying to talk about an insight myself here, and I am not sure I have succeeded, but I feel that what I want to say is this: the simple is analogous to the true.
1. I do not wish to go any further with this and make references to authors who not all of us study here, such as Wittgenstein, for instance, who has an attractive approach to these kind of questions, which for him, very roughly, are not legitimate questions, but ‘questions’ that arise from a certain misunderstanding of the roughly, are not legitimate questions, but ‘questions’ that arise from a certain misunderstanding of the distinction between words of different uses. He calls words such as time, knowledge, meaning, etc. “odd-job” words to which we mistakenly apply the same form of questioning as we do to “what is a dog?” to which the answer is unequivocally: This!” (by pointing out to a dog). If I’m wrong about this, however, help me see it, because I’m not sure so far that talking about “the simple” is any different from talking about “the virtuous”.
2. Is this unintelligible, a stretch, or do you see what I mean?
3.A propos, I was struck by your house in this respect earlier today; how natural and comfortable it felt moving around such a pleasant little house where everything so simple and unassuming was where it belonged—I don’t know how I could answer: how do you know/learn where things belong?--as opposed to the uncomfortable feeling I remember getting a year and a half earlier in your trailer, trying to use the restroom and even just moving around. Not only how we arrange our furniture, but how we even built our houses is indicative of a tendency to order and simplicity.
4. This may look like an unaccounted for assumption, but I am thinking of it as something obvious, something that is proved daily, and not just in the laboratory...
5. Does this make sense? I’m not sure how to explain it