The Unreasonable Effectiveness Of Mathematics In The Natural Sciences - A Possible Explanation
The Unreasonable Effectiveness Of Mathematics In The Natural Sciences [Eugene Wigner (b. 1902 d.1995)]
[quote=Eugene Wigner]The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.[/quote]
[quote = wikpedia]Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is [u]something bordering on the mysterious and that there is no rational explanation for it[/quote]
It seems that scientists and mathematicians are puzzled by the "unreasonable effectiveness" of math in the natural sciences. In other words, they have no explanation why math has proved to be such a useful tool in science, allowing scientists to derive precise mathematical formulae that describe almost every conceivable phenomena except maybe women :grin:
Anyway, I will attempt to provide an explanation for why math is such a fine tool in describing our universe.
Imagine two universes A and B with two objects each and governed by two different laws. For simplicity the laws will be about motion only but the idea is applicable, mutatis mutandis, to all laws in any universe.
Universe A:
1. Two objects
2. Law of motion: if one object is struck by another object the struck object will move. This law is non-mathematical
Universe B:
1. Two objects
2. Law of motion: if one object travelling at velocity w, strikes another object at an angle x the, the struck object shall move with a velocity y at an angle z., This law is mathematical
As must already be obvious, universe A is chaotic as the struck object can assume any velocity and any path - the motion is completely random as the law fails to fully describe motion.
The situation is different in universe B as there is order - the struck object's velocity and path is fully determined by the object striking it. Motion is fully described in this universe and there can be no chaos in the sense that the struck object can assume any velocity or any trajectory.
As I mentioned before, this is true for any law, whether physical or chemical - the law governing such phenomena has to be mathematical for there to be order.
Can life arise in chaos, in universe A? Considering that life requires different sets of chemical and physical laws acting in concert and the formation and persistence of some pattern of matter and energy which is impossible in a chaotic universe that is completely random, it follows that life is not possible in universe A.
This isn't the case for universe B though; it has order and allows patterns in matter and energy to take hold and persist, thereby making life possible.
To make the long story short, math gives rise to order and order gives rise to life and life gives rise to Eugene Wigner. The reason why math is so effective in describing the universe we live in is that life can't exist in a nonmathematical universe of chaos. Every universe life exists must have order and where there's order there'll always be mathematics.
Comments...
ADDENDUM: As @Harry Hindu was kind enough to point out, there is a pattern in universe A viz. an object will move when struck. So, to call a system that has a law, a pattern, chaotic is a contradiction (or so it seems).
Indeed, chaos precludes any and all patterns and universe A has a pattern (if struck, move). So, then is universe A not chaotic? To answer "yes" amounts to saying universe A has order. If the law/pattern in universe A (if struck, move) is just that and nothing more then, struck objects can assume any velocity or trajectory in a random fashion, the net effect of which would be chaotic/patternless motion. What does this, this patternless motion, imply but that motion in universe A is chaotic?
It seems, prima facie, that we're faced with a contradiction. We agree that universe A has a pattern (if struck, move) and, at the same time, universe A doesn't have a pattern (struck objects take on random trajectories and velocities).
How do we make sense of this, if there's any sense at all in it?
In knowing that in universe A, a struck object will be in motion, we have a pattern, but in not knowing how that motion will occur, we have no pattern.
Perhaps if look at the usage of the word "pattern", we'll make some progress. A pattern is found in something - patterns don't exist without a thing it's a characteristic of. So, the pattern "if struck, move" is found in the physical interaction of objects in universe A and B. This pattern is labeled motion. The pattern "if an object with velocity w strikes an object at an angle x then the object struck shall move at a velocity y, at an angle z" is a pattern, a mathematical pattern, in motion in universe B and not in A. In other words we're looking at patterns in patterns.
As might be already obvious to you now, there is no contradiction in the universe A scenario because the chaos in it (random velocities and trajectories of struck objects) is the absence of a mathematical pattern in the nonmathematical pattern "if struck, move"; in simple words there is no pattern in the pattern. The chaos (mathematical) is at a different "level" than the pattern (nonmathematical). I believe I already mentioned this is one my comments.
Coming back to what I really want to demonstrate here, the fact that there is chaos in a universe like A with nonmathematical laws, and that life is about patterns (order) in matter-energy, it follows that a nonmathematical universe wouldn't be able to support life. A universe with life has to be mathematical.
[quote=Eugene Wigner]The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.[/quote]
[quote = wikpedia]Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is [u]something bordering on the mysterious and that there is no rational explanation for it[/quote]
It seems that scientists and mathematicians are puzzled by the "unreasonable effectiveness" of math in the natural sciences. In other words, they have no explanation why math has proved to be such a useful tool in science, allowing scientists to derive precise mathematical formulae that describe almost every conceivable phenomena except maybe women :grin:
Anyway, I will attempt to provide an explanation for why math is such a fine tool in describing our universe.
Imagine two universes A and B with two objects each and governed by two different laws. For simplicity the laws will be about motion only but the idea is applicable, mutatis mutandis, to all laws in any universe.
Universe A:
1. Two objects
2. Law of motion: if one object is struck by another object the struck object will move. This law is non-mathematical
Universe B:
1. Two objects
2. Law of motion: if one object travelling at velocity w, strikes another object at an angle x the, the struck object shall move with a velocity y at an angle z., This law is mathematical
As must already be obvious, universe A is chaotic as the struck object can assume any velocity and any path - the motion is completely random as the law fails to fully describe motion.
The situation is different in universe B as there is order - the struck object's velocity and path is fully determined by the object striking it. Motion is fully described in this universe and there can be no chaos in the sense that the struck object can assume any velocity or any trajectory.
As I mentioned before, this is true for any law, whether physical or chemical - the law governing such phenomena has to be mathematical for there to be order.
Can life arise in chaos, in universe A? Considering that life requires different sets of chemical and physical laws acting in concert and the formation and persistence of some pattern of matter and energy which is impossible in a chaotic universe that is completely random, it follows that life is not possible in universe A.
This isn't the case for universe B though; it has order and allows patterns in matter and energy to take hold and persist, thereby making life possible.
To make the long story short, math gives rise to order and order gives rise to life and life gives rise to Eugene Wigner. The reason why math is so effective in describing the universe we live in is that life can't exist in a nonmathematical universe of chaos. Every universe life exists must have order and where there's order there'll always be mathematics.
Comments...
ADDENDUM: As @Harry Hindu was kind enough to point out, there is a pattern in universe A viz. an object will move when struck. So, to call a system that has a law, a pattern, chaotic is a contradiction (or so it seems).
Indeed, chaos precludes any and all patterns and universe A has a pattern (if struck, move). So, then is universe A not chaotic? To answer "yes" amounts to saying universe A has order. If the law/pattern in universe A (if struck, move) is just that and nothing more then, struck objects can assume any velocity or trajectory in a random fashion, the net effect of which would be chaotic/patternless motion. What does this, this patternless motion, imply but that motion in universe A is chaotic?
It seems, prima facie, that we're faced with a contradiction. We agree that universe A has a pattern (if struck, move) and, at the same time, universe A doesn't have a pattern (struck objects take on random trajectories and velocities).
How do we make sense of this, if there's any sense at all in it?
In knowing that in universe A, a struck object will be in motion, we have a pattern, but in not knowing how that motion will occur, we have no pattern.
Perhaps if look at the usage of the word "pattern", we'll make some progress. A pattern is found in something - patterns don't exist without a thing it's a characteristic of. So, the pattern "if struck, move" is found in the physical interaction of objects in universe A and B. This pattern is labeled motion. The pattern "if an object with velocity w strikes an object at an angle x then the object struck shall move at a velocity y, at an angle z" is a pattern, a mathematical pattern, in motion in universe B and not in A. In other words we're looking at patterns in patterns.
As might be already obvious to you now, there is no contradiction in the universe A scenario because the chaos in it (random velocities and trajectories of struck objects) is the absence of a mathematical pattern in the nonmathematical pattern "if struck, move"; in simple words there is no pattern in the pattern. The chaos (mathematical) is at a different "level" than the pattern (nonmathematical). I believe I already mentioned this is one my comments.
Coming back to what I really want to demonstrate here, the fact that there is chaos in a universe like A with nonmathematical laws, and that life is about patterns (order) in matter-energy, it follows that a nonmathematical universe wouldn't be able to support life. A universe with life has to be mathematical.
Comments (47)
I can see Tegmark's Mathematical Universe as an outlandish, but somewhat plausible argument, but I don't see a reasonable path to a conclusion here.
Quoting TheMadFool
Huh. Is that true? Does order always imply mathematics? A Pharaoh decrees orders that govern the treatment of slaves - does this orderliness imply mathematics?
Does chaos imply no mathematics? I think it possible to envision a chaotic universe that is entirely mathematical since mathematics underlies chaos theory in our own universe. I can easily produce a chaotic structure by simple iteration. Sensitive dependence on initial conditions (Butterfly Effect) does the trick.
Order¹: the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method.
Order²: A command
Quoting jgill
Mathematical chaos theory is, to my reckoning, simply about unexpected complexity in fully deterministic systems. Nevertheless, there are patterns that can be discerned. The chaos I'm reffering to is completely devoid of any pattern and is also wholly random and undeterministic except perhaps in a qualitative sense.
Mathematics is simply a method of symbolizing relationships. F=m*a. We can use words or numbers to represent the relationship, which is what you did here. Neither one is chaotic if the both contain LAWS. You said the same thing, one with words and the other, with more symbolic details about the relationship. The extra detail allows me to use the representation to make predictions about other instances of where objects bump into each other and what the results will be.
Well, my contention is that a nonmathematical law leads to chaos but a mathematical law leads to order. Your comment reveals a not unexpected bias engendered by (over)exposure to the laws of this universe in which we live where all known phenomena are non-chaotic. You've been conditioned to associate "law" with "no chaos" as there are no known exceptions that could've made you think otherwise.
The fact of the matter is a [s]qualitative [/s]nonquantitative/nonmathematical law can exist such as the one I described in my OP that can only lead to chaos; on the other hand, a quantitative/mathematical will always lead to order.
You are wrong in your assessment of my worldview. It seems like you didn't really bother reading what I said. You are the one that labeled each statement in each universe as a LAW, not me, and then you went on about explaining what is the case:
Quoting TheMadFool
If it were a universe of chaos, then it wouldn't always be the case that if one object is struck by another object the struck object will move. Maybe the objects would pass right through each other. Maybe the struck object doesn't move, and you wouldn't be able to make statements like this, which means that there aren't any laws.
A law is a statement, or a representation, of some state-of-affairs. You can represent things with words or with numbers and numbers are just words. (7 is seven, 8 is eight, f=m*a is force equals mass times acceleration).
How did you learn to pronounce, or even say, "7" if not for "seven"? Just as Dr. is an abbreviation of doctor, 7 is an abbreviation of seven.
So all you have done is provide the same explanation in both universes, just with different symbols. Both statements allow you to make predictions - that struck objects will move. How can you make predictions in a chaotic universe?
But chaos has an inherent order, as non-linear dynamics clearly establishes through the use of attractors.
If we speak about a universe where there exist 3 masses that follow the classical laws of gravitation, you have the 3-body Problem which is mathematical. But if you imagine the universe of a cell (I'm not going to pretend I understand how the cell works at even a high school level), you can define all the parts and describe how all those parts interrelate, and even though this may be a consistent and determinable system of relationships, we wouldn't generally consider that mathematical. Yet, we can go on to consider something like Conway's Game of Life, which is similar to a cell in the sense that you describe the parts (the grid of cells, "live" and "dead" states, neighborhood, tick) and how those parts interrelate (the rules by which cells are born and die according to their neighborhood and their propagation through ticks), but this is mathematical in nature. Which brings me back to my initial question: what makes one thing mathematical and the other not?
I suspect the devil's in the details. The cell is defined and described in non-quantitative means, by empirical observation. Something about this is different from how we define and describe the Game of Life, which is a priori, in a sense. I don't think it's merely quantitative though, as plenty of mathematics isn't quantitative, in the sense that not all mathematics is about numbers and measurement. But if it's not quantitative, what is it?
Remember that I had to find an explanation for why the laws of the universe are mathematical. Thus the necessity for a scenario with two different universes, one operating under a mathematical law and the other under a nonmathematical one. Only then could I demonstrate why s universe with life has to be mathematical.
As far as I'm concerned, regarding your claim that the imagined universe A with the nonquantitative law is not chaotic, all I ask from you is to describe the pattern (since you deny this is chaos) in the motion of objects with the law: if one object is struck by another object the struck object will move. The pattern you might claim exists is simply that the struck object will move but how this motion occurs will be totally chaotic, no? It is this chaos I'm referring to. All other physical and chemical laws too will result in chaos if the laws that govern them are nonmathematical for the same reason. This chaos, I hope we're clear on what I mean, is sufficient to prevent any kind of order in universe A. Without order, life, which is simply patterns (order) of energy and matter, is impossible.
That said, you're not entirely wrong in saying that the existence of a law implies there is no chaos; all I can say about that is this chaos is at a different level than the chaos I'm referring to.
Agreed! Particularly since mathematics confers no Darwinian biological advantages!
No, that both universes are explainable. Like I said, you can use words or numbers to explain it, and numbers are just words.
Quoting QuixoticAgnostic
As I said, mathematics describes the relationship between things. If things are interrelated then they are explainable in mathematical terms (which are just words). F=m*a is the symbolic representation of how force, mass and acceleration are interrelated.
Quoting QuixoticAgnostic
It seems to me that it is all quantitative. Look at TheMadFool's Laws, they both include the concept of one, and another which is quantitative.
Quoting TheMadFool
Objects of perception are bounded and quantitative by their very nature. It is how the mind conceives of the world that isn't made up of objects, but of relationships.
I don't know. It's your imaginary universe. You tell me.
If one object strikes another and the other will move is a law, it seems to be an incomplete law because it doesn't tell how the struck object will move after being struck, or anything about the nature of the objects themselves. If there isn't a pattern then it can be said that the universe is chaotic. It wouldn't be chaotic because one pattern is explained by mathematics and the other is explained by words, which I pointed out isn't a difference because numbers are just words. It would be chaotic because you couldn't establish a pattern of like causes leading to similar effects.
The distinction is in relationships that don't merely describe, but define. For example, in Conway's Game of Life, you can say there is a dead cell with 3 live cells in its neighborhood. I just described a relationship between a cell and its neighbors, but this isn't a mathematical description because it doesn't tell me about how these things interact with one another. If I say, however, that cells are initially either "live" or "dead" and transition states according to rules X, Y, Z, then I am making a mathematical statement because it is defining the behavior and interactivity of things.
Although what I'm confused about is you say both universes are explainable, but deny that the first universe is mathematical? You say "if things are interrelated, then they are explainable in mathematical terms", so in universe A the two objects are interrelated, so they are explainable in mathematical terms, which means the universe is mathematical, no? Regardless, we're both in agreement that there is no fundamental difference between universe A and B.
Well, if I could upload imagery of a dynamical system I devised I would challenge you to find a pattern.
Quoting TheMadFool
As a former meteorologist I disagree. Your approach is very Newtonian, which is not bad, but rather insufficient.
Your Universe A is a fairy tale and thus cannot be compared to our universe. (My late Hungarian ex-father-in-law carried on a lengthy correspondence with Wigner. I'm surprised the Nobel Laureate did not formulate and advance your argument. :roll: )
What's wrong with imagined scenarios? They're legit philosophical devices, no? Isolate the key variable and do something to it and see what follows and so on...
Yes. Exactly. Nonmathematical laws are incomplete and also precludes order, a necessary ingredient for life.
This last paragraph is irrelevant so long as you agree that nonmathematical laws come off as incomplete. In this incompleteness is the seed for chaos and where there is chaos, life, but a pattern (order) in matter-energy, becomes impossible.
Quoting jgill
I used motion as a jumping board because it's conveniently simple and still to the point. Also, I mentioned quite clearly that the basic idea - that nonmathematical laws lead to chaos - applies to all the laws of science.
By the wat what would be a non-Newtonian approach? How would that affect my argument?
Albert Einstein used thought experiments (a lot). That should allay your concerns, hopefully.
Quoting jgill
Hi, Mr. Meteorologist. Wonderful to have you on the forum. Chaos must be familiar to you; after all Edward Lorenz who discovered The Butterfly Effect was a meteorlogist too.
I agree that Chaos Theory is a legitimate field of study in mathematics. I don't contest that. However, correct me if I'm wrong, mathematical chaos is deterministic chaos i.e. chaotic systems of Chaos Theory are technically not chaotic - they are simply extremely complex but completely deterministic systems. Wittgenstein may have an issue with the use of the term "chaos" here.
The chaos I'm referring to is a different kind of chaos - one that is non-deterministic due to the complete lack of mathematics in the laws of a universe; such a universe would be chaotic too and this particular type of chaos would preclude life, if life is construed as patterns (order) in matter-energy.
I'm surprised too. :chin: My argument is very simple. Nonmathematical laws lead to chaos and chaos doesn't allow life. Ergo, a universe that has life will always have mathematical laws.
Sure it's a mathematical description. You just described the number of dead cells relative to live cells. TheMadFool did the same thing in his OP, that I pointed out in my previous post to you, but you seem to have missed it.
The former and the latter are simply answering, or explaining, two different things. They each answer two different questions, yet they both use mathematics in their explanation. They also use words, and I have said that numbers are words, so essentially you explain everything with words, which mathematics consists of.
Quoting QuixoticAgnostic
I never denied that the first universe was mathematical. I specifically said that it was, and even bolded the text to make it easy for you to see, but you still missed for some reason. I also said that both universes are explainable and by explainable I mean that you use words to represent some state-of-affairs, and mathematical explanations consist of words.
So, if numbers are words, and you explain things with words, then there is no distinction between a universe that you can explain with mathematics and one you can explain with words.
You asked me a question about the state of your imaginary universe. If it is imaginary, then that means we can make up whatever we want, so you could imagine that your Universe A actually does have more information than what some statement (law) about it exhibits.
We don't live in an imaginary universe. We live in a universe that is a certain way and that we come into, being ignorant of. We make stuff up in our minds, but the universe has a different story that contradicts our story about it, so we have to adjust our story from time to time to fit our observations of it. The universe fine tunes our explanations via our observations of it. To keep making stuff up about the state of the universe without using our observations to filter the stuff we make up would be a symptom of delusional disorder.
Quoting TheMadFool
As I have said many times already, there is no distinction between mathematical and nonmathematical laws. Mathematical explanations are worded explanations.
Both you and QuixoticAgnostic have used numbers in both of your "non-mathematical" laws!
Quoting TheMadFool
No, explanations are incomplete or complete depending on the question being asked. What kinds of questions can we ask about both universes? Can we ask the same questions about both? If so, would we receive the same answers? Why or why not?
Hmm. . .
Quoting Harry Hindu
Yeah, that certainly doesn't look like denial :roll:
Also, reducing mathematics to numbers and numbers to words (not even descriptive words, but literally just the denotation "one", "two", "three", etc.) doesn't make a whole lot of sense. Mathematics encompasses much more than just quantity and words don't always describe mathematical relationships. Your "numbers are words" argument is confused: just because numbers are words doesn't mean all words are numbers, which means it isn't necessarily the case that words describe mathematical relationships.
"You can say there is a dead cell with 3 live cells in its neighborhood" is a description of a state, but not a mathematical law because it doesn't describe the behavior of the system or how things interact, which was the point of me bringing that up. "However, I would qualify that mathematics isn't just about relationships, but interrelationships or, better put, interactivity. The distinction is in relationships that don't merely describe, but define."
This is the part I was referring to:
Quoting Harry Hindu
The part you are referring to was me translating "mathematical" to "explainable", as mathematics is a type of explanation - so no contradiction.
The type of explanation depends on what type of question is being asked.
Quoting TheMadFool
TheMadFool's assertion was that Universe A is chaotic because the Law isn't a mathematical explanation, while Universe B is orderly because the Law is a mathematical explanation.
But each explanation is about, or answers, very different questions about each Universe.
Order and chaos are not dependent on what type of questions we ask. They are dependent on whether or not a patterns can be found and then used to make predictions in each Universe.
Quoting TheMadFool
Chaos is a term than can refer to randomness or complexity. Randomness can be a result of our ignorance. Once we are able to establish a predictable pattern in the Universe, randomness disappears and becomes order.
In Universe A, you established that a pattern exists - that a struck object moves when struck. If you want to say that the way in which it moves is random, that's fine, but Universe A is still not completely random, as you were still able to establish a pattern within it, and that pattern answers a different question than what your pattern within Universe B does.
If you're saying that the question we are asking about Universe B is unanswerable in Universe A, why is it unanswerable - because we just haven't established a pattern yet, or that there is no pattern to establish? How would you know which is the case in Universe A, unless you imagined it because after all, it is an imagined Universe, and you are its god. Unfortunately, we aren't god in this Universe.
The fact is that, in this universe we have established patterns - a great many of them, and each of them answers different types of questions - mathematical and non-mathematical.
Once upon a time, to think humans could fly was delusional. Imagination, which you seem to have a dim view of, is what has made heavier-than-air flight possible.
Quoting Harry Hindu
I'm curious about this statement. How so?
This amounts to the contradiction that:
mathematical is nonmathematical
Perhaps, being exposed to only mathematical laws of our universe, you find it difficult to conceive of a world with nonmathematical laws.
If I maybe so bold as to give you a glimpse of nonmathematical laws in this/our universe and how it leads to chaos, consider social laws. One good example is the law: a good turn deserves another. You wouldn't dispute the fact of this law being nonmathematical would you? If you do, set it aside for the moment and just go with me on this. If there's no math involved at all in this law then the interaction among people following this law would be erratic [chaotic], no? There would be no pattern, i.e. no order, in the behavior of people following this law. No predictions would be possible.
However, the moment quantification/math enters the fray, patterns will begin to emerge on the basis of who did more/less/equal (math) good to whom. The law I spoke of becomes: a (more/less/equal) turn deserves another (more/less/equal turn). Of course, the quantification/math involved is not precise but all that's important for you to note is this: even a crude and imperfect quantification generates patterns (order) and so you can imagine what a completely mathematical version of a simple nonmathematical law can achieve.[b][/b]
Agreed.Quoting Harry Hindu
Agreed but the pattern is not sufficient to prevent chaos or more accurately the pattern is not sufficient to generate the order necessary for life.
So you seem to be contradicting yourself in using two different qualifiers for chaos - one that the universe has unpredictable patterns and the other being the universe can't be explained in mathematical terms.
Now you are saying that chaos something that can't be prevented. How does it follow that because you can answer a mathematical question with a mathematical answer that that prevents chaos?
It seems to me that chaos was prevented the moment you established any pattern in a universe. From that point you would just be talking aboutQuoting TheMadFool, and not randomness.
Compare the following three:
1. Universe with no laws. There would be absolutely no pattern. This is the chaos you're talking about. I agree with you here.
2. Universe with a nonmathematical law. As you rightly pointed out, there is a pattern that the law describes but does this pattern preclude chaos?
Quoting Harry Hindu
Why did you say "an incomplete law" and use the word "how"? For the simple reason that the nomathematical law "if an object strikes another object, the object struck should move" is not sufficient to predict the behavior of objects in a universe that has such a nonmathematical law. Why can't you predict? That would be because there is no pattern in the motion of objects in such a universe. Where there is no pattern, there is chaos no? Basically, there is a pattern in that struck objects will move but there is no pattern in how the struck object will behave/move.
In the scenarios I put for consideration the pattern you see is nonmathematical but the chaos is mathematical. I'm not contradicting myself. Why is the pattern nonmathematical? Why, numbers don't figure in it. Why is the chaos mathematical? There's no pattern in the trajectory or speed, both mathematical entities, of objects.
3. Universe with mathematical laws. These laws, you will agree, are both "complete" and permits us to know "how" matter-energy interact. There's no room for chaos in such a universe because everthing that happens to matter-energy will evince a pattern.
So life, being patterns of matter-energy, isn't possible in a universe with either absolutely no laws or nonmathematical laws as these are chaotic universes.
Quoting Harry Hindu
Please read the above.
Actually if mathematics is the theory of everything, then objects consist of the laws of nature. Then as laws unto themselves objects exhibit freedom. Objects anticipate their alternative future states for some parameters.
And without randomness / decisionmaking then it is impossible to generate information.
Definition:
one:
1. denoting a particular item of a pair or number of items
2. the lowest cardinal number; half of two; 1
https://www.researchgate.net/publication/287422786_The_myth_of_the_clockwork_universe_Newton_newtonianism_and_the_enlightenment
:nerd:
Again, you aren't using a consistent definition of chaos. Is chaos a lack of patterns or a lack of mathematical explanations? As and I have pointed out, one doesn't necessarily entail the other.
Even then, in your Universe A, you have both patterns and mathematical explanations.
Quoting TheMadFool
Quoting TheMadFool
Seems quantitative to me.
Quoting TheMadFool
In Universe A, why does an object move when struck?
To say that there is a pattern of objects moving when struck, yet no pattern in how the struck object moves, seems to be an incomplete explanation. When explaining why an object moves when struck, you'd be explaining the nature of the objects themselves. Is there something in the nature of the objects that would inform us how they would move when struck? Again, what kinds of questions can we ask about the objects in Universe A? It seems to me that we can ask both non-mathematical and mathematical questions about the objects in Universe A, as in how many objects exist, and how many objects it takes to move another.
I agree completely with you that chaos is the absence/lack of patterns. We'll begin from there.
Let's take an example of our universe: look at the physics of falling bodies. Observation informs us that when an object is let go in the air, they fall to the earth. This is the nonmathematical pattern we see in our universe and is called free fall. The next step is, as any science history buff will tell you, to find out if this nonmathematical pattern (free fall) has, within it, a mathematical/quantitative pattern. In other words, we're looking for a pattern (a mathematical one) in a pattern (a nonmathematical one). This is the story of Galileo that every science student knows.
As you know, Galileo discovered that, for falling objects, the distance travelled by a falling object is directly proportional to the square of the time spent falling.
Can you see here that a pattern can appear in two places? This should be obvious to you since you did mention in a previous post that one of my imagined universes lacks a description of "how" motion takes place. Anyway, here's what the situation regarding patterns vs chaos with Galileo is:
1. There is a pattern, a nonmathematical one, in the way the earth interacts with objects that are let go in the air - they always fall. This pattern is called free fall. Call this pattern X.
2. There is another pattern, a mathematical one, in free fall as discovered by Galileo: the distance an object falls is directly proportional to the square of the time spent in falling. Call this pattern Y.
You can see, quite clearly, pattern Y is a pattern in pattern X and pattern X is NOT pattern Y. Keep this in mind.
Now, since pattern X is not pattern Y, it is completely possible for pattern X to be present (no chaos = free fall ) and pattern Y to be absent (chaos = there is no mathematical relationship between the distance an object falls and the time it takes to fall) without contradiction for one is a pattern and the other is the pattern in that pattern - two entirely different things . This is exactly what's the case with universe A which has the nonmathematical pattern, "if struck, move", which you were so kind to point out to me (thanks), but lacks the mathematical pattern "if one object moving at velocity w, at an angle x, strikes another object, the struck object will move at a velocity y at an angle z." Let's call the pattern "if struck, move", pattern/law S. In universe A, pattern/law S exists but what's missing is a mathematical pattern in the pattern/law S. I'm using the term "chaos" consistently and without contradiction.
Quoting Harry Hindu
So, a lexicographer was so intent on causing confusion that he repeated the same meaning twice and even numbered them differently as 1 and 2?
Another thing I wanted to mention. Here's another law you're familiar with: if "one" object of mass m (kg) has an acceleration of a (m/s^2) then, the force acting on it is F = m * a. Does the "one" in the sentence that states the law make the law quantitative? By your logic "yes" but as a matter of fact the correct answer is "no". Similarly the "one" in the universe A's law, "if one object is struck by another object, the struck object will move" doesn't make the law quantitative.
I like thought experiments a lot. But your Universe A is too much to wrap my mind around. Sorry.
Thought experiments can't get simpler than the one I came up with in this thread. I know I'm not right, at least not completely right; if it were this simple, Eugene Wigner, a proven genius, would've been onto it before I could've said Jack Robinson.
Ok.
Quoting TheMadFool
Yes, but is the fact that a pattern exists at all in Universe A indicative that Universe A is not chaotic?
How much chaos vs. order has to exist for a Universe to qualify as chaotic?
Quoting TheMadFool
Yes, but is the fact that you can refer to a number of objects in Universe A and how many are part of the process that we are talking about (falling bodies, and a falling body has to be falling relative to something else) indicative that Universe A is quantitative?
How much non-math vs. math has to exist for a Universe to qualify as non-mathematical?
Quoting TheMadFool
So now you are using the kind of chaos that you said you aren't using - complexity.
Chaos theory Quoting wikipedia
The initial condition is free fall. Can an object fall at a different rate if the initial condition of free fall wasn't met?
People talk to each other - this is a pattern. However they maybe talking gibberish - this is chaos. I can't explain this any better. You'll have to reread my post and come back with better points.
Quoting Harry Hindu
So, "one" in if "one" object of mass m (kg) has an acceleration of a (m/s^2) then the force acting on it F = m * a is central to the quantitative nature of the statement?
Quoting Harry Hindu
I'm using your definition of "chaos". Does that include complexity?
Well, it is simple. Too simple. Not enough information to begin an intelligent discussion, IMO. Underlying any approach is an agreement of just what math is. Is there mathematics without symbols, words, or figures? In Universe A, Motion=F(Impact), a qualitative mathematical statement. Impact produces motion. Sorry, it all seems a little flaky.
First, I'd like to call attention to the law of the conservation of energy. I'll argue that the law implies that the physical world is describable mathematically and I'll also argue that violation of the law requires that something can come from nothing and also that something can vanish into nothing - features of the universe that, for thousands of years have been considered illogical, though I understand that some of our quantum mechanics experts now have reason to believe that the ex nihilo fit principle is not exactly what we took it to be.
When an object strikes another object, if chaos results, that is to say, if the two objects can bounce off in any directions at any speeds, then there must be a net gain or loss of energy in the interaction. If that is so, then if we give up the rule that mathematics must be observed in physics, we also must give up the rule that energy is conserved. That seems to me to be a lot to give up. One wonders if there could even be a universe in which that is possible.
I can't prove it but I'm inclined to believe that "our" universe is mathematically describable, not because of the anthropomorphic principle, but because any universe - even any universe in an "expanding multiverse" in which physical constants may be different, must be so describable. It is the nature of existence that that be so. If it's not, then I suspect that logic and reason will go out the window along with mathematics. Or perhaps I should say, there couldn't even be a window for anything to go out.
Alan
Quoting jgill
:chin:
You're as fickle as the weather, Mr. Meteorologist.
I don't see how you could be saying that people are talking to each other if they are "talking" gibberish. This is a contradiction. You need to come back with better arguments.
Quoting TheMadFool
Sure, because it's not two or three, etc. It's basically saying:
(n * mass) * a = F
where n = the number of objects.
Quoting TheMadFool
I thought we agreed to use this definition:
Quoting TheMadFool
but then you went about using a different definition - the one that refers to complexity, which I showed. Maybe you should be the one that re-reads posts, not me.
[quote=Cambridge Dictionary]I was so nervous, I just started talking gibberish.[/quote]
Quoting Harry Hindu
:rofl: So, had I said, "if two objects strike each other and then the one, identified as p, having a mass m moves at an acceleration a" then the force acting on p is F = 2 * m * a?
Quoting Harry Hindu
You made a good point and I responded to it.
Too much of a stretch. But nice rejoinder! :cool:
I don't see what's so funny, unless it's your silly imaginary universe you're laughing at. Is your imaginary universe funny?
If two objects strike one object, what happens?
Do all the objects involved have mass?
Are any objects moving? If so, at what velocity. Remember that velocity is relative.
Are these not all mathematical questions that I'm asking you about the objects in your universe that you claim is non-mathematical? And these questions need to be answered before you even get to F=m*a
If the objects you are talking about are coherent objects the space within and around them must be coherent and laws would simply be a description of the shape of this space. Once you have space you have law/mathematics.
I think your argument is right. As far as I can see, there is no possible alternative reality in which mathematics cannot describe the physical characteristics of the universe. Any such alternative would be unstable and beyond description. The effectiveness of mathematics in the natural sciences isn't "unreasonable". It is ineffectiveness of mathematics that would be unreasonable.
Is it true that "Chaos is chaotic beyond imagining"? I think it is though we might wish to rephrase the argument to avoid the criticism that it's only because we lack imagination. What I'd like to say is that the laws of mathematics aren't as separate from the laws of physics as some of us might imagine. To say that one apple plus one apple = two apples isn't just about numbers. It's true of apples, and also oranges, houses, words, and all other discrete objects. Apples are countable. Their masses and velocities are measurable. They don't wink in and out of existence. Their mathematical properties are as much a part of their being as their mass, their extension (i.e. occupation of space), their color, their taste, and everything else about them.
If countability, addition of velocities, and other descriptions of apples did not follow mathematical laws, then the very most basic principles of our physics, and the most fundamental descriptions of our universe, would not obtain - including the law of conservation of energy. We would not exist either since we too are countable and subject to gravity, additions of velocity, and so on. Your statement about this that "... Once you have space you have law/mathematics." seems to me to be right on target.
I think the argument that you and I (if I understand you correctly) are propounding is intuitive but it probably needs a real philosopher/mathematician/physicist beyond my limited knowledge to properly flesh it out.
Alan