quanta'namo nay! Posted February 1, 2015 Share Posted February 1, 2015 I've recently stumbled on some people claiming that some mathematical equations can produce random results. To me this seemed quite strange. Essentially, to my understanding mathematical equations are always by definition deterministic and this results in the possibility to make predictions in terms of science. Granted, my understanding often needs revising and I enjoy doing that upon encountering solid arguments. That means I am learning and my mind is evolving. It seems to me that if science is to use mathematics to approximate the behaviour of the Universe and if that modelling is successful, that in turn indicates that the Universe is entirely deterministic. There is of course the notion of quantum mechanics and "randomly existing virtual particles". One question that arises here is how could something that is random be possibly modelled by entirely deterministic means. However, I think the main question here I want to present to all mathematicians: How would you respond to someone claiming that a mathematical equation can produce random results? And additionally: Would you consider successful mathematical modelling of the universe as indication of the universe being deterministic? Link to comment Share on other sites More sharing options...
studiot Posted February 1, 2015 Share Posted February 1, 2015 Would you consider successful mathematical modelling of the universe as indication of the universe being deterministic? No. What about do we know of any successful mathematical modelling of the universe? Again No. How would you respond to someone claiming that a mathematical equation can produce random results? You need to understand what is meant by an equals sign in mathematics. It often indicates the result of a process (Chemists for instance used to use it for this but now more often use an arrow). That process may or may not produce a predeterministic result. Sometimes an equation may have an infinite number of valid solutions, so any one chosen at radom will hold true. Sometimes we say 'an equation cannot be solved in closed form'. Does that make it deterministic? Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 1, 2015 Author Share Posted February 1, 2015 What I meant was that any mathematical equation f(x) when given input X and resulting in output Y, will always result in output Y with the input X. In that sense all mathematical equations are deterministic. If they were not, no mathematical proof could be made at all. I suppose the requirement of mathematical proofs is deterministic equations. An equation may have multiple solutions, but they're not random but deterministic as they always hold. Link to comment Share on other sites More sharing options...
studiot Posted February 1, 2015 Share Posted February 1, 2015 If you think that an equation will is deterministic what is the output of this equation? x + (-x) = 0 So what is x? Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 1, 2015 Author Share Posted February 1, 2015 (edited) If you think that an equation will is deterministic what is the output of this equation? x + (-x) = 0 So what is x? x is the input. Any given input will deterministically produce the output. Just that your equation will reduce to 0 = 0 and takes no input... Edited February 1, 2015 by quanta'namo nay! 1 Link to comment Share on other sites More sharing options...
Strange Posted February 1, 2015 Share Posted February 1, 2015 It seems to me that if science is to use mathematics to approximate the behaviour of the Universe and if that modelling is successful, that in turn indicates that the Universe is entirely deterministic. In quantum theory, for example, mathematics is used to make predictions based on probability. In other words, the interactions are not deterministic and all we can do is predict the probability of different outcomes. And then there are chaotic systems which are completely deterministic but unpredictable. This includes the weather but also simple things like a double pendulum. Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 #5 quanta'namo nay! x is the input. Any given input will deterministically produce the output. So are you saying I can input anything I like to an equation and get an output? Please note I mean valid mathematical objects, like numbers not cats, dogs etc, I am not being facetious. Link to comment Share on other sites More sharing options...
ajb Posted February 2, 2015 Share Posted February 2, 2015 What I meant was that any mathematical equation f(x) when given input X and resulting in output Y, will always result in output Y with the input X. This is closer to the notion of a function, that is a many-to-one relation on two sets. You are discounting mappings that are many-to-many, including one-to-many. Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 In quantum theory, for example, mathematics is used to make predictions based on probability. In other words, the interactions are not deterministic and all we can do is predict the probability of different outcomes. And then there are chaotic systems which are completely deterministic but unpredictable. This includes the weather but also simple things like a double pendulum. Probabilistic calculations are also essentially deterministic in their mathematical form. If virtual particles were an accurate description of reality they would also need to be deterministic as they are said to form the subatomic processes, given that the operation is not random and unpredictable. I'd say that if something is random it can not follow any process that can be modelled (i.e. is predictable). So are you saying I can input anything I like to an equation and get an output? Please note I mean valid mathematical objects, like numbers not cats, dogs etc, I am not being facetious. I suppose the inputs to a mathematical equation must be other equations as a substitutions or numeric values. Mathematics as the language only deals with equations and numeric values, physics as an application of that language then adds concepts like types (e.g. cats and dogs). This is closer to the notion of a function, that is a many-to-one relation on two sets. You are discounting mappings that are many-to-many, including one-to-many. An equation (or function) may have multiple inputs and the output may be an equation or a numeric value. If you use sets then you are talking about multiple instances of usage of the function I suppose. If you have a different view point to this I'd ask you to present a simple example with some simple set(s). Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 studiot, on 02 Feb 2015 - 12:41 AM, said: So are you saying I can input anything I like to an equation and get an output? Please note I mean valid mathematical objects, like numbers not cats, dogs etc, I am not being facetious. I suppose the inputs to a mathematical equation must be other equations as a substitutions or numeric values. Mathematics as the language only deals with equations and numeric values, physics as an application of that language then adds concepts like types (e.g. cats and dogs). So, just to confirm, are you saying I can freely input any number to an equation and the output is always deterministic? Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 So, just to confirm, are you saying I can freely input any number to an equation and the output is always deterministic? In the sense of repeatability yes. Always when you input the same input to the same function it will deterministically give you the same result. Assuming no errors were made Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 So if I choose x = 8 as the input to this equation, what is the output ? x2 - 4 = 0 Your theory is also in difficulty with simple probability. I can state the probability of heads in a fair coin toss is 0.5. X = the probability of heads, output 0.5 Deterministic as you say. But what is the output of the equation X = the result of next toss of the coin? How is that deterministic? Link to comment Share on other sites More sharing options...
Strange Posted February 2, 2015 Share Posted February 2, 2015 So if I choose x = 8 as the input to this equation, what is the output ? x2 - 4 = 0 You have moved the goalposts. You started with an equation where x could take any value, when quanta'namo nay! correctly pointed out that x could take any value in that equation, you changed the equation. I don't know what point you are trying to make, but you are doing a poor job of it! Probabilistic calculations are also essentially deterministic in their mathematical form. True: whenever you calculate the probability you will get the same result for the probability. But that doesn't tell you what will actually happen. If virtual particles were an accurate description of reality they would also need to be deterministic as they are said to form the subatomic processes, given that the operation is not random and unpredictable. I'd say that if something is random it can not follow any process that can be modelled (i.e. is predictable). Sorry, but quantum processes are (*) random and unpredictable. The best we can do is predict the probability of certain outcomes. (*) According to our current best theories and the evidence available so far, etc. etc. Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 So if I choose x = 8 as the input to this equation, what is the output ? x2 - 4 = 0 You have already specified that you want a specific output (0). If I translate this into words of spoken language you get: What input gives you an output of zero when squared and subtracted four? The deterministic answer is of course +-2. Determinism ensures that your answer to this question is always the same. True: whenever you calculate the probability you will get the same result for the probability. But that doesn't tell you what will actually happen. Sorry, but quantum processes are (*) random and unpredictable. The best we can do is predict the probability of certain outcomes. (*) According to our current best theories and the evidence available so far, etc. etc. If something is random then the frequencies of occurrence (probabilities) are also random and not constant. Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 No so. You are the one who said that X is the input and confirmed that X could be any number. The problem is that you cannot obtain an output although you can indeed write the equation 82 - 4 = 0 Which is what i actually said, written out as an equation. If it is of interest this is because the equals sign in this equation is differnt from the equals sign in the first one I presented. The equation, x + (-x) = 0 is an identity ; That is it holds for all values of X The equation X2 - 4 = 0 is an equality. Perhaps you have heard of and understand the difference? Secondly you have not answered my second example in post#12 Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 No so. You are the one who said that X is the input and confirmed that X could be any number. The problem is that you cannot obtain an output although you can indeed write the equation 82 - 4 = 0 Which is what i actually said, written out as an equation. If it is of interest this is because the equals sign in this equation is differnt from the equals sign in the first one I presented. The equation, x + (-x) = 0 is an identity ; That is it holds for all values of X The equation X2 - 4 = 0 is an equality. Perhaps you have heard of and understand the difference? Secondly you have not answered my second example in post#12 If you say "A equals B", this can only have two possible outputs. They are TRUE or FALSE. The input possibilities are infinite. If you say "What does A equal to", this has an infinite amount of output possibilities depending on an infinite amount of input possibilities. Do you see the difference? Link to comment Share on other sites More sharing options...
Strange Posted February 2, 2015 Share Posted February 2, 2015 If something is random then the frequencies of occurrence (probabilities) are also random and not constant. The probabilities may be constant and predictable. For example, the probability of rolling a 6 or the probability of an atom decaying in a given time. But you still can't say which atom will decay or when. You are the one who said that X is the input and confirmed that X could be any number. He didn't say that was so for any function. You are just being silly, now. Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 The probabilities may be constant and predictable. For example, the probability of rolling a 6 or the probability of an atom decaying in a given time. But you still can't say which atom will decay or when. Rolling dice is known to be deterministic. We can change the shape of the dice and their resulted probabilities will change. Why would you say virtual particles are random? Link to comment Share on other sites More sharing options...
swansont Posted February 2, 2015 Share Posted February 2, 2015 Rolling dice is known to be deterministic. We can change the shape of the dice and their resulted probabilities will change. They are still probabilities, though, which was the point. The outcome of rolling dice is probabilistic. Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 If you say "A equals B", this can only have two possible outputs. They are TRUE or FALSE. Not so by Godel's Theorem. You still haven't answered my second point in post#12 Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 They are still probabilities, though, which was the point. The outcome of rolling dice is probabilistic. It seems you are saying that what is deterministic is also probabilistic and vice versa. You can get probabilities for pretty much anything. Wiki says: "Probability is the measure of the likeliness that an event will occur", so absolutely everything is probabilistic; even something with 0 probability. What do you mean when you say something is "probabilistic"? Not so by Godel's Theorem. You still haven't answered my second point in post#12 Mathematics is a language of high precision. The latter part of your post #12 is something I do not understand. To me it seems gibberish, but that may be just my lack of understanding of your notation. Perhaps you could elaborate and make the equation precise? Link to comment Share on other sites More sharing options...
studiot Posted February 2, 2015 Share Posted February 2, 2015 Studiot post#12 point no.2 Your theory is also in difficulty with simple probability. I can state the probability of heads in a fair coin toss is 0.5. X = the probability of heads, output 0.5 Deterministic as you say. But what is the output of the equation X = the result of next toss of the coin? How is that deterministic? The latter part of your post #12 is something I do not understand. To me it seems gibberish, but that may be just my lack of understanding of your notation. Perhaps you could elaborate and make the equation precise? I'm sorry if you did not understand this. It is basically the same point both strange and swansont have made. There is a deterministic equation that will yield a definite probability P(E) of some event E. Hence there is another definite probability of a different event = {1-(P(E)}. Now you started this thread with the proposition Essentially, to my understanding mathematical equations are always by definition deterministic and this results in the possibility to make predictions in terms of science. So, given P(E) or {1-P(E)} we are all asking what is the deterministic prediction of the outcome? Link to comment Share on other sites More sharing options...
Strange Posted February 2, 2015 Share Posted February 2, 2015 Rolling dice is known to be deterministic. That is true. They are chaotic, which means that even though they behave deterministically they are unpredictable. So all you can do is predict probabilities. Why would you say virtual particles are random? Quantum theory is inherently random, probabilistic and (sometimes) acausal. Even though it can be described (very accurately) by mathematics. Link to comment Share on other sites More sharing options...
quanta'namo nay! Posted February 2, 2015 Author Share Posted February 2, 2015 I'm sorry if you did not understand this. It is basically the same point both strange and swansont have made. There is a deterministic equation that will yield a definite probability P(E) of some event E. Hence there is another definite probability of a different event = {1-(P(E)}. Now you started this thread with the proposition So, given P(E) or {1-P(E)} we are all asking what is the deterministic prediction of the outcome? Ah, ok. So if your input to the equation is just probabilities so will your output be. I suppose that is what people do when a deterministic model is too complex for them to handle. People want things easy Of course you can produce an equation (or a set of them) to model a deterministic coin toss. Then input will just be quite many and the equation quite complex. Here's an example of a deterministic model of tossing something more complex than just a coin: http://en.wikipedia.org/wiki/Stair_Dismount The models are so complex that it is not feasible to post as forum messages. Take a look at some of the videos of the above simulation, they're hilarious Link to comment Share on other sites More sharing options...
swansont Posted February 2, 2015 Share Posted February 2, 2015 It seems you are saying that what is deterministic is also probabilistic and vice versa. You can get probabilities for pretty much anything. Wiki says: "Probability is the measure of the likeliness that an event will occur", so absolutely everything is probabilistic; even something with 0 probability. What do you mean when you say something is "probabilistic"? studiot just answered this. What do you mean by deterministic? Link to comment Share on other sites More sharing options...
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