• Landoma1
    38
    I think all participants here know about the statement of the unreasonable effectiveness of mathematics. Shouldn't we, rather, speak of it's reasonable effectiveness? I can't see nothing unreasonable about it and can't even imagine how else it could be.

    Let's not forget though where we apply it. To dead Nature. In the human realm it seems unreasonable if effective indeed.
  • Relativist
    1.8k
    Why is math effective? Because there is structure to the world that is describable with mathematics.

    Why is the world describable with mathematics? Because there are regular, consistent physical relations between objects that have an inherent mathematical component (like an inverse square law).

    Why are these relations present? They just are. We're they not, we would not be here to question.

    Why is it more reasonable to expect an absence of such relations?
  • Jackson
    1.8k
    To dead Nature.Landoma1

    Nature is not dead.
  • Joshs
    4.2k
    Why is math effective? Because there is structure to the world that is describable with mathematics.

    Why is the world describable with mathematics? Because there are regular, consistent physical relations between objects that have an inherent mathematical component (like an inverse square law).
    Relativist

    I was with you in your first paragraph. But the fact that there is structure to the world does not mean that the world comes to our awareness packaged an ‘inherent’ way that is already mathematical. Nature became mathematizable when we contributed our own peculiar interpretive structures to it.
    As you can see, I’m a mathematical constructivist, not a platonist.
  • Jackson
    1.8k
    I think all participants here know about the statement of the unreasonable effectiveness of mathematics.Landoma1

    Never heard of it before you.
  • Joshs
    4.2k
    Never heard of it before you.Jackson

    look up Eugene Wigner.
  • Jackson
    1.8k
    look up Eugene Wigner.Joshs

    Why?
  • Joshs
    4.2k


    Why?Jackson

    Why not?
  • Jackson
    1.8k
    Why not?Joshs

    Better things to do.
  • Joshs
    4.2k
    Better things to do.Jackson

    Name one
  • T Clark
    10.8k
    the fact that there is structure to the world does not mean that the world comes to our awareness packaged an ‘inherent’ way that is already mathematical. Nature became mathematizable when we contributed our own peculiar interpretive structures to it.Joshs

    This is a good way of putting it.
  • Gnomon
    2.8k
    I think all participants here know about the statement of the unreasonable effectiveness of mathematics. Shouldn't we, rather, speak of it's reasonable effectiveness? I can't see nothing unreasonable about it and can't even imagine how else it could be.Landoma1
    I'm guessing that Wigner's use of "unreasonable" was ironic or tongue-in-cheek. In view of the randomness & uncertainty of its Quantum foundation, it is perhaps surprising that on the Macro level of reality, its structure & processes are predictable & consistent. In other words, there is an underlying logic to the order of reality. And mathematics is simply an abstract form of Logic.

    Moreover, Logic is essential to the extraction of meaningful information by humans (Reason). Some might say that Human Logic & Natural Logic both result from the Natural Laws that caused the Big Bang to self-organize into the smoothly functioning mechanism we see today. That orderly structure of interrelationships is what allows human mathematics (logical inference) to be both Reasonable and Effective. But why should a random & accidental "explosion" (expansion) of something from almost nothing turn out to be lawful (orderly & organized)? Perhaps Wigner saw signs of design in the world, but chose to comment on it equivocally, for professional reasons. :cool:
  • Relativist
    1.8k
    I was with you in your first paragraph. But the fact that there is structure to the world does not mean that the world comes to our awareness packaged an ‘inherent’ way that is already mathematical. Nature became mathematizable when we contributed our own peculiar interpretive structures to it.Joshs
    No, it's not packaged in an inherent way, but the success of our inferred mathematical relations suggests there is an ontological basis to it.

    As you can see, I’m a mathematical constructivist, not a platonist.
    I'm also not a Platonist. I have an Aristotelian view of immanent universals (more directly: an Armstrongian view).
  • RussellA
    678
    I think all participants here know about the statement of the unreasonable effectiveness of mathematics. Shouldn't we, rather, speak of it's reasonable effectiveness?Landoma1

    The effectiveness of mathematics is neither reasonable nor unreasonable.

    I observe the world. I observe that all things being equal, what happened in the past will happen in the future. This is neither reasonable nor unreasonable, it is just a fact about the world. I observe on the table in front of me my pen touching my pencil, and observe that the mere fact of touching does not cause a change of velocity of either my pen or my pencil. I discover facts about the world by observing the world. These facts are neither reasonable nor unreasonable, they are just how the world is.

    "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner.

    A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass. This was contrary to what Aristotle had taught: that heavy objects fall faster than lighter ones.

    Richard Hamming reflected on and extended Wigner's Unreasonable Effectiveness in 1980. Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Suppose that a falling body broke into two pieces. Of course the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tie the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?

    Humans can invent many different mathematical systems. Those mathematical systems that are found to correspond with the world consistently through time are kept, otherwise they may be discarded. That some mathematical systems are discovered to be more effective that others is neither reasonable nor unreasonable, it is just what is.

    Who is to say that the mere fact of my pen touching my pencil does not result in a change of velocity is a reasonable or unreasonable thing to happen ?
  • Joshs
    4.2k
    No, it's not packaged in an inherent way, but the success of our inferred mathematical relations suggests there is an ontological basis to it.

    “As you can see, I’m a mathematical constructivist, not a platonist.”(Josh)

    I'm also not a Platonist. I have an Aristotelian view of immanent universals (more directly: an Armstrongian view).
    Relativist

    What I reject is the idea that the regularity and consistency of physical relations reduces to differences of degree that are not at the same time differences in kind. Put differently, quantitative measurement introduces qualitative change at every repetition of the counting.
  • Relativist
    1.8k
    What I reject is the idea that the regularity and consistency of physical relations reduces to differences of degree that are not at the same time differences in kind.
    Put differently, quantitative measurement introduces qualitative change at every repetition of the counting.
    Joshs
    I don't follow you, but I'll elaborate on my view: laws of nature are relations between kinds of things. Kinds are universals, and laws of nature are universals. This is the metaphysical theory of law realists.

    Your first sentence sounds consistent with law realism. I don't know what to make of your second sentence, other than that it sounds like an interpretation of quantum mechanics. Please explain.

    Are you a nominalist?
  • Joshs
    4.2k
    Your first sentence sounds consistent with law realism. I don't know what to make of your second sentence, other than that it sounds like an interpretation of quantum mechanics. Please explain.

    Are you a nominalist?
    Relativist

    I’m a phenomenologist, but the inextricable relation between quantitative interaction and qualitative
    transformation I described comes from Deleuze , whose touchstone was Nietzsche. For both orientations lawfulness , self-identity, the ability to carve out and iterate a pristine quantitative realm within a qualitative dimension are idealizations that invent rather than represent the real. A ‘kind’ is not a category, object, identity. It is a differentiation. There are no quantities within kinds. Every iteration of quantity is a change of kind.
    Strange stuff from a realist perspective.
  • Relativist
    1.8k
    A ‘kind’ is not a category, object, identity. It is a differentiation. There are no quantities within kinds.Joshs
    Isn't "electron" a kind? Do they not all have an electric charge of quantity -1?
  • Joshs
    4.2k
    Isn't "electron" a kind? Do they not all have an electric charge of quantity -1?Relativist

    Look at the period at the end of this sentence. Now keep on staring at it. We say that the period is a kind, an identity persisting in time with attributes and properties that belong to it. Mathematics begins from , and depends on , such reifications. But , most fundamentally , that is not how you are experiencing the period as you continue to gaze at it. It is not simply that your gaze or body posture subtly shifts your perspective, but that your sense of the meaning of what you are perceiving also shifts is subtle ways every moment. Each repetition of the period is a subtly new interpretation of it. This is not even including the ways in which the period , as a natural object, is not simply a system of relations among fixed kinds of physical particles. When we employ concepts like ‘kind’,property’ and ‘attribute’ so as to see electrons with numerically assigned charge, we are masking all of this underlying subtle but incessant dynamism and change for the sake of convenience. Our mathematics begins only after we have concealed what happens within ‘kinds’.
  • Relativist
    1.8k
    A period is a fuzzy concept. It could mean a small, physical mark, of no specific shape, a set of pixels; an abstract concept, a word that English speakers interpret as a semantic clue. I'm only reifying it if I treat it as an abstract object that exists in the world. I assure you, I don't.

    The problems with "period" aren't present for electron. I regard an electron as a type of ontic object, -specifically, objects with a certain set of properties (such as -1 electric charge, a specific rest mass, etc). I gather you disagree, so I'd like to understand your point of view.
  • 180 Proof
    11k
    Nature is not dead.Jackson
    Nature is undead.
    .
    look up Eugene Wigner.
    — Joshs

    Why?
    Jackson
    As you say, you've never heard of (Wigner's paper) "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and you should start doing your own homework so that you can contribute intelligently to the current discussion.

    As you can see, I’m a mathematical constructivist, not a platonist.Joshs
    :up:

    As an embodied cognitionist (Lakoff, Dehaene), "the effectiveness of mathematics" seems quite reasonable to me as well.
  • jgill
    2.7k
    Look at the period at the end of this sentence. Now keep on staring at it.Joshs

    Look into this box of apples. They are all Delicious apples, a kind of apple. Now look closely at each one after carefully counting them - there are 24. Each apple is unique, being distinguished from the others in small ways. We see this as we contemplate these apples, a particular kind of apple. After a bit each apple seems to turn its best side toward our gaze, and we begin to contemplate what may lie on their opposite sides. In so doing we drift into a meditative state in which apples prevail, even those not Delicious.

    Our mathematics begins only after we have concealed what happens within ‘kinds’.Joshs

    :chin:
  • Banno
    19.9k
    :up:

    It's the same as being amazed that the word "apple" is so well-suited to our talk of apples...
  • Wayfarer
    16.8k
    I think all participants here know about the statement of the unreasonable effectiveness of mathematics. Shouldn't we, rather, speak of it's reasonable effectiveness?Landoma1

    You can't be sure they will, but just in case, here is the essay you're referring to.

    Why Wigner says it is 'unreasonable' is because of the sense in which mathematical conjectures sometimes produce completely unforseeen predictions which turn out to be true - that 'mathematical concepts turn up in entirely unexpected connections', as Wigner says.

    Wigner gives some examples, but admittedly they are difficult to understand unless you have some background - after all, Wigner was a Nobel laureate in mathematical physics, for discoveries derived from mathematical symettries, which I presume few here would be familiar with. But that said, one of the examples he gives is this:

    This (case) originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory.

    However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say "if the mechanics as here proposed should already be correct in its essential traits." As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom.

    The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we "got something out" of the equations that we did not put in.
    — Eugene Wigner, Unreasonable Effectiveness...

    I take it that this means that the equations in question made predictions which were not even contemplated in relation to the original problems they were supposed to solve. And there have been other such cases in the history of science. Take for example Paul Dirac, another Nobel laureate in physics:

    The father of antimatter was the remarkable English physicist Paul Dirac (1902-1984), considered by many to be the greatest British theorist since Sir Isaac Newton.

    His research marked the first time something never before seen in nature was “predicted” – that is, postulated to exist based on theoretical rather than experimental evidence. His discovery was guided by the human imagination, and arcane mathematics.

    For his achievement Dirac was awarded the Nobel prize for physics in 1933 at the age of 31.

    Read more: https://www.newscientist.com/article/dn17111-how-dirac-predicted-antimatter/
  • litewave
    793
    Let's not forget though where we apply it. To dead Nature. In the human realm it seems unreasonable if effective indeed.Landoma1

    Why is math effective? Because there is structure to the world that is describable with mathematics.Relativist

    As soon as there are ANY differences in the world, you have a structure describable by mathematics. You can count the differences, you can make combinations of the differences, you can make combinations of those combinations, you can order the combinations (for example by size). The whole known mathematics is reducible to set theory, which is basically a theory of combinations (sets are combinations of their members).
  • RussellA
    678
    Thought experiments and the "unreasonable" nature of mathematics

    Aristotle taught that heavy objects fall faster than lighter ones.

    However, Richard Hamming's thought experiment gives a powerful reason why heavy objects should fall at the same speed as lighter ones. This thought experiment allows us to predict that universally a heavy object should fall at the same speed as a lighter one, not only on Earth today, but on the far side of the universe and millions of years from now. Such thought experiments can predict both what cannot be seen and what cannot be experimentally foreseen.

    Personally, I have consistently observed across space and time that the mere fact of my pen touching my pencil does not result in a change of velocity of either of them, from which I may reasonably agree with Hamming that it follows that there is a universal law that heavy objects should fall at the same speed as lighter ones.

    Does not such a prediction show the "unreasonable" power of reason, and by extension logic and mathematics ?

    Beliefs are always self-referential, and therefore always well-suited to the world and always "unreasonably" effective

    For example, a table is the relation between a table top and table legs. The same applies to apples, governments, unicorns, houses, ethics, etc. As relations don't ontologically exist in the world (FH Bradly) but only in the mind (the Binding Problem), tables don't ontologically exist in the world but only in the mind. As the concept of tables only exists in our thoughts and talk, it should be no surprise about the "unreasonable" effectiveness of our thoughts and talk about tables, as thoughts about tables and tables are one and the same thing.

    The question is, are our laws of nature comparable with the situation as regarding the table? As tables only exist in the mind, perhaps our laws of nature only exist in our mind, and consequently, as both self-referential, obviously both well-suited to the world and "unreasonable" effective.

    My deduction of the universal law that heavy objects should fall at the same speed as lighter ones is only valid as long as there is a condition of satisfaction between my beliefs and the state of affairs that obtains in the world. My belief clearly does not determine the state of affairs in the world, but as my belief is self-referential, my belief is both well-suited and "unreasonably" effective as to what I believe to be the state of affairs in the world.

    Paradigm shifts in my beliefs does not alter the self-referential nature of belief

    However, my belief in such a law of nature, my belief in how the world is structured, is no guarantee that what I observe will continue to comply with my present beliefs, in which case I may be forced through the same kind of paradigm shift as described by Thomas Kuhn and be forced to develop a new set of beliefs that correspond with my new experiences and observations.

    The point remains that even this new set of beliefs will also be self-referential, in that even my new beliefs will be well-suited and "unreasonably" effective as to what I believe to be the state of affairs in the world.

    Our self-referential belief in reason, logic and mathematics can only ever be well-suited to the world and "unreasonably" effective

    IE, regardless of what beliefs I may have, what reasoning. logic or mathematics I use, my beliefs will always be well-suited and "unreasonably" effective to my understanding of the world around me because of the self-referential nature of belief.
  • Joshs
    4.2k


    Look into this box of apples. They are all Delicious apples, a kind of apple. Now look closely at each one after carefully counting them - there are 24. Each apple is unique, being distinguished from the others in small ways. We see this as we contemplate these apples, a particular kind of apple. After a bit each apple seems to turn its best side toward our gaze, and we begin to contemplate what may lie on their opposite sides. In so doing we drift into a meditative state in which apples prevail, even those not Delicious.jgill

    In this example , the category ‘apple’ subsumes the particularities of the individual apples. The parts
    can vary ( kinds of apples) without altering the whole ( the category apple). But does this logical subsumption bear any resemblance to how we actually construct and experience the relation between parts and wholes? Or is it the case, as the Gestaltists say, that the whole precedes its parts and the parts redefine the whole?

    Wittgenstein analyzed the issue of the meaning of parts and wholes.

    “There is a tendency rooted in our usual forms of expres- sion to think that the man who has learned to understand a general term, say, the term "leaf', has thereby come to possess a kind of general picture of a leaf, as opposed to pictures of particular leaves. He was shown different leaves when he learned the meaning of the word 'leaf'; and showing him the particular leaves was only a means to the end of producing 'in him' an idea which we imagine to be some kind of general image. We say that he sees what is common to all these leaves; and this is true if we mean that he can on being asked tell us certain features or properties which they have in common. But we are inclined to think that the general idea of a leaf is something like a visual image, but one which contains what is common to all leaves. This again is connected with the idea that the meaning of a word is an image, or a thing correlated to the word. (This roughly means, we are looking at words as though they all were proper names, and we then confuse the bearer of a name with the meaning of the name).

    (d) Our craving for generality has another main source: our preoccupation with the method of science.I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer questions in the way science does. This tendency is the real source of meta- physics, and leads the philosopher into complete darkness.”
    (The Blue Book, pp. 17)
  • RussellA
    678
    Parts may exist in the world, but the whole only exists in the mind
    The whole is a set of parts, but even a part is a set of parts. For the sake of argument, treat the parts as elementary and logical rather than real. As relations have no ontological existence in the world, but only in the mind, the parts may exist in the world, but the whole only exists in the mind.

    From particular observations to general laws
    From my particular observations in different locations over a period of time that the mere fact that my pen touches my pencil does not result in a change of velocity of either, I can make the general assumption that two objects in contact does not result in the change of velocity of either. I can also make the general assumption about the regularity of the laws of nature. This confirms Richard Hamming"s thought experiment that universally, heavy bodies fall at the same speed as lighter ones, contrary to Aristotle's teaching.

    That I can predict a heavy object and a lighter one will fall at the same speed not only on Earth today but on the far side of the universe millions of years from now does initially seem to illustrate the unreasonable power of reason, and by extension logic and mathematics.

    General laws only exist in the mind
    As general laws, such as the law of nature that two objects when touching does not cause a change of velocity of either require relations between parts, a relation between object A and object B, then general laws can only exist in the mind and not the external world.

    We have a whole concept of a universal table (apples, government, ethics, etc) in our minds, which is based on the observation of the relationship of particular parts in our external world. We have the whole concept of a universal law of nature in our minds, which is based on the observation of the relationship of particular parts in our external world. Both the concept of universal table and universal law of nature extend to the far side of the universe. Does our ability to extend the law of nature to the far side of the universe show the unreasonable power of reason ? No, no more than our extending the concept of a table to the far side of the universe shows the unreasonable power of reason.

    Concepts such as tables and the law of nature remain in the mind. It does not follow that our projection of these concepts onto the external world makes them states of affairs that obtain in the external world. The fact that we can project our concept of tables and laws of nature onto the far side of the universe cannot be said to show the unreasonable power of reason, as neither of these concepts actually obtain as states of affairs.

    As reason, logic and mathematics are self-referential, they cannot be said to be either reasonably or unreasonably effective
    Our beliefs are self-referential, in that my concept of a table or a law of nature is of necessity well-suited to what I observe in the external world. IE, on the one hand I have a belief in my mind of a table or law of nature existing in the external world. On the other hand a table or law of nature is a belief that exists only in the mind and not the external world.
  • Landoma1
    38
    Never heard of it before you.Jackson

    It was Eugene Wigner who spoke of the "unreasonable effectiveness of math". Nature has dead and alive elements. Many deed phenomena (which doesn't mean they don't contain at least the seeds of life) behave in fixed patterns, contrary to living phenomena. For example, the principle of least action applies to dead matter but not to life.
  • Landoma1
    38
    Why Wigner says it is 'unreasonable' is because of the sense in which mathematical conjectures sometimes produce completely unforseeen predictions which turn out to be trueWayfarer

    I still don't see why that's unreasonable. It seems only reasonable if What's so unreasonable about getting things out you didn't put in, as the anti-particle?
  • Landoma1
    38
    As soon as there are ANY differences in the world, you have a structure describable by mathematicslitewave

    Astute observation. But how would you describe my face changing from neutral to laughing in math? The principle of least action applies to falling stones but does it apply to a bacteria?
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