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   Does math represent truth?
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Mickey1
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Does math represent truth?  
« on: Mar 18th, 2010, 8:03am »
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The quotes and references below are from my comments – also mentioned elsewhere  here http://goneri.nuc.berkeley.edu/pages2009/slides/Jensen_Comments%20to%20t he%20students.pdf  They are in part a comment on Alison Macfarlane’s presentation on the US high level waste problems in a summer course in Berkeley last August: The Science and Policy of Nuclear Waste Disposal in the US: http://goneri.nuc.berkeley.edu/pages2009/slides/Macfarlane-1.pdf    
Alison Macfarlane is a member of the US Energy Department’s blue ribbon Commission on nuclear waste disposal, a hot topic in the US after your President dumped the plans for a Yucca Mountain repository.
 
Now you have the background. Here we go:
The answer is no. However, even if it did we would have trouble formulating the truth-models:
 
"A scientific theory is represented in a language and can never completely represent
reality. There is always a distance between the signifier and the signified and a
translation from the object into a system of signs or code, even when the most
elementary observations are involved" (Nolin 1993).  We believe Newtonian mechanics can  “predict” the solar system’s behaviour (even without relativistic corrections) almost a million years into the future, but it is still the Sun that rises, not Newtonian mechanics.  
 
Another even more disturbing example – since this theory is more widely confused with
reality – is offered by the interpretation of results from quantum mechanics. Audi (1973)
explains that “To say that the results / from the uncertainty relations/ are inherent in the nature
of measurement is to lend our present theories a timeless quantity which, considering the
nature of theories, they could not possess”. Michael Audi is associated with what is sometime
called a minority view on quantum mechanics in that he denies the need for Bohr’s duality
principle, interpreting with Born and others all phenomena in particle terms, but the quote
above is a more general observation.
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rmsgrey
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Re: Does math represent truth?  
« Reply #1 on: Mar 18th, 2010, 10:28am »
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Depends what you mean by "truth".
 
Maths is unusual among human activities in that it offers absolute certainty, but with no guarantee of relevance.
 
On the other hand, much of maths arises from abstracting patterns from reality, so, at the least, applying it offers a good description of that aspect of reality. What's been described as maths' "unreasonable effectiveness" in describing (aspects of) reality makes it natural to speculate that there's something deeper there...
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Mickey1
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Re: Does math represent truth?  
« Reply #2 on: Mar 19th, 2010, 7:37am »
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Math can offer certainty in the sense that a mathematical proof offers a permutation of the original axioms.
 
In terms of connection to reality this is close to circular  reasoning.  
 
Perhaps we might discuss the issue first here:  
 
1 Does (or can) mathematical proofs offer more than a permutation  of stated definitions and axioms?  
 
Next to come:
2 Can reality blend with formulars - models - used to describe it so that it is not possible to distinguish between the models (math) and the reality?  
 
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Re: Does math represent truth?  
« Reply #3 on: Mar 19th, 2010, 7:57am »
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on Mar 19th, 2010, 7:37am, Mickey1 wrote:
Math can offer certainty in the sense that a mathematical proof offers a permutation of the original axioms.
A mathematical proof offers more than just a random permutation of axioms. In fact a mere permutation of axioms cannot be a proof. A proof is characterized by inference from axioms to theorems. And inference rules are not axioms.
 
And it does tell you things you did not know. I don't know where Jupiter will be in twenty years, even if I know the laws of gravity and the position and speed of Jupiter, I still do not know where it will be in twenty years, until I actually calculate it. I do not have at my command all knowledge that can be inferred from the knowledge I have. Not until I go through the process of inferring it.
Mathematics (including logic, though logicians may prefer the reverse), gives a means for valid inference. And in that sense gives certainty that the conclusions you reach based on the knowledge you have is valid (under certain given assumptions about the input; as they say, garbage in garbage out).
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Re: Does math represent truth?  
« Reply #4 on: Mar 19th, 2010, 9:46am »
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on Mar 19th, 2010, 7:57am, towr wrote:
And it does tell you things you did not know. I don't know where Jupiter will be in twenty years, even if I know the laws of gravity and the position and speed of Jupiter, I still do not know where it will be in twenty years, until I actually calculate it.

 
Even then, you only know where Jupiter will be in twenty years if it should move according to the laws of gravity as you know them (and there are no significant effects from previously unknown masses...)
 
As you said, GIGO...
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Re: Does math represent truth?  
« Reply #5 on: Mar 21st, 2010, 6:26am »
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I think that no Math cannot represent "TRUTH" as a whole, however, with that said, I do think that Math, and subsequent mathematical equations can represent  A TRUTH.
 
Take the golden ratio for example, here is a number for instance, that can be found repeated in nature.  Well-known artists, architects, and several others have used this number as a means for creating an aesthetically pleasing look in their artistic and architectural creations.
 
The Golden Ratio is considered to be an irrational number because it cannot be expressed as a fraction, and yet you find it everywhere in nature.
 
And it is well known that the Golden Ratio has a relationship to the Fibonacci sequence, which is a number patterning exponential growth.
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Re: Does math represent truth?  
« Reply #6 on: Mar 21st, 2010, 7:41am »
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on Mar 21st, 2010, 6:26am, WildwindE wrote:
The Golden Ratio is considered to be an irrational number because it cannot be expressed as a fraction, and yet you find it everywhere in nature.
I find the "and yet" somewhat strange. As if "rationality" of numbers is supposed to say anything about whether they occur in nature. Pi is irrational, sqrt(2) (diagonal of a unit square) is irrational, in fact one can safely say most numbers in nature are irrational.  
 
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And it is well known that the Golden Ratio has a relationship to the Fibonacci sequence, which is a number patterning exponential growth.
An infinite number of sequences "pattern" exponential growth. Any second order recurrence relation that has positive coefficients greater than 1, and two positive starting values, grows exponentially. (And that hardly characterizes them all.)
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Mickey1
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Re: Does math represent truth?  
« Reply #7 on: Mar 22nd, 2010, 4:11am »
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Where will Jupiter be in 10 billion years? The Sun could turn into a red giant and swallow the inner planets, the solar system could be visited by heavy celestial objects and we might be less confident that present calculations could find the answer. However, this is not my major counterargument, which is instead that calculation is different from observation. Only if man (or some other creature) in a distant future was capable of changing nor only his environment but also the physical constants (velocity of light etc) would I accept that reality and math were indistinguishable, and we would be living in Matrix-type universe (keeping in mind that even Matrix had consistency problems).
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Mickey1
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Re: Does math represent truth?  
« Reply #8 on: Mar 22nd, 2010, 4:14am »
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PS
Actually, the advanced theorem is less "truthful" in my sense  than its building blocks who have some connection to the truth. While simple numerical relations may be accepted as truth also about real scenarios involving collections of objects ("natural" numbers and perhaps also "natural" relations), I can not see the Riemann conjecture as having any meaningful relation to any “real” scenario.
« Last Edit: Mar 22nd, 2010, 4:21am by Mickey1 » IP Logged
towr
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Re: Does math represent truth?  
« Reply #9 on: Mar 22nd, 2010, 4:55am »
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on Mar 22nd, 2010, 4:11am, Mickey1 wrote:
However, this is not my major counterargument, which is instead that calculation is different from observation.
That's not a counter-argument to what I said. What I'm saying is that inference can reveal new knowledge; and therefore is more than just rearrangement of things you already knew (be they axioms or observations or anything else).
 
But on the topic of observation; observation really only tells you 'what is' at a fleeting moment in time. You can observe that a car is 100 meters from you at one moment, 50 meters the next, and neither observation will tell you that that the following moment you'd better be somewhere else. You have to infer that truth (if it is true). Without superimposing a model -- that there are objects that are moving, in what way they move, etc -- you miss out a lot of the universe.
Mathematical model simply fit extremely well, and allow you to extract a lot more 'truth' from observation. It's the kind of thing that will allow us to predict whether observed asteroid will hit us or not, and whether we should try to deflect them if we want them not to hit us. You wouldn't have any knowledge of that kind without mathematical models. To what extent theories are "true" in an absolute sense is not really as important as how accurate the predictions are.  
For all you know we are brains in a vat, and all observations we make are merely delusions put into our brains by a virtual reality machine. That's a truth you could never find out, and one that's completely irrelevant to any practical approach to living your life.
« Last Edit: Mar 22nd, 2010, 5:00am by towr » IP Logged

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Mickey1
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Re: Does math represent truth?  
« Reply #10 on: Mar 23rd, 2010, 10:50am »
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I have a different point of view. It is exactly the absolute sense I am referring to. I agree to all the rest i.e. the usefulness of models, except the Berkelian view that we cannot know if the world exists.  That is an axiom, lying deeper, similar to the axiom that the world was not created 20 minutes ago, discussed elsewhere in this forum, under creation vs. evolution.
Insurance people will tell you that the car is not always where you project it to be. In your (Anglo- Sachsen) legal tradition this is reflected in the difference between witness statements (where the car was) and expert opinion (where the car can be projected to be in the future).
The problem is more obvious in a more complicated context such as the one I come from: Post-close safety assessments, dealing with question: where is the waste's content of radionuclides in a million years from now? The models are there but not always the confidence. Many (especially US) engineers are overconfident in their models. But there is no difference in principle with waste and the car.
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Re: Does math represent truth?  
« Reply #11 on: Mar 23rd, 2010, 11:37am »
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We don't have an Anglo-Saxon legal tradition where I am. And in any case, witnesses are notoriously unreliable.
 
I'm not really sure what you're trying to say with the rest of your post. Are you claiming that there is a way to arrive at absolute truths? Such as, perhaps, that the world we perceive is in fact as we perceive it to be, and not some delusion?
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Mickey1
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Re: Does math represent truth?  
« Reply #12 on: Mar 23rd, 2010, 1:01pm »
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Yes
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Re: Does math represent truth?  
« Reply #13 on: Mar 23rd, 2010, 1:10pm »
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Well, please share how. Because I really don't see it myself.
The only truths I can discern are all conditional on assumptions; such as that our senses and observations do not deceive us (even though we know they can: just take optical illusions as an example, or dreams).
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Mickey1
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Re: Does math represent truth?  
« Reply #14 on: Mar 26th, 2010, 1:26am »
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The world is approximately how we perceive it to be but a stricter version is that the world exists independent of our minds, i.e. whether we perceive it correctly or not.
 
If you were dreaming reading this – or in another type of delusion - you would still have to follow the same cause of action (to quote SF satirist Robert Sheckley) and you would perhaps follow the simplest explanation, to find that the world has an independent existence. If you then wake up to an objective reality there is no harm done.
 
Let me summarize my experiences here (I will be digitally away for some time):
1 I should have presented my definition of truth in the beginning
2 I realise that simple number relations can (also) be truthful statements about classes of objects in an absolute way – but hardly complicated theorems.
3 Engineering models, including Newton's, are never truth, but are useful since reality can be described well enough for many purposes by models.
4 These philosophical observations about truth are not intended in any way to diminish the value of modern (post Galileo) science, only to put them in perspective and to prepare for their future development.
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Re: Does math represent truth?  
« Reply #15 on: Mar 26th, 2010, 5:01am »
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on Mar 26th, 2010, 1:26am, Mickey1 wrote:
The world is approximately how we perceive it to be but a stricter version is that the world exists independent of our minds, i.e. whether we perceive it correctly or not.
But that doesn't help us to know anything about this independently existing world. Because our 'knowledge' is in our minds, and the world is independent of it.
I still don't see the point in worrying about 'absolute truth'. It exists, but we can't know more about it than that.
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Re: Does math represent truth?  
« Reply #16 on: May 23rd, 2010, 3:26am »
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That does really depend on your definition of "the truth". The thing is math is one of the things that is what was, what is and what will always be. And then using "math" is how people rate accuracy.
 
It's the basis of everything. How you are right or wrong. It's the truth in terms of quantifying things.
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Mickey1
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Re: Does math represent truth?  
« Reply #17 on: Mar 22nd, 2012, 2:35pm »
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Here is a quote from Dr Math on the internet:
 
"There are those (called Platonists) who believe that all mathematical concepts and objects exist and are, as it were, waiting in the wings to be discovered. Others believe that mathematics is the creation of persons. For most of us, these are not burning issues.  However, our "general" graduate education in  mathematics tends towards the latter viewpoint, at least operationally."
 
It appears that math here (in a sense) is closer to politics than anything else.
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Re: Does math represent truth?  
« Reply #18 on: Mar 22nd, 2012, 11:20pm »
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I don't see in what way mathematics is remotely like politics in any sense, other than that it's something people do.
Saying maths is a creation of people, while true in a sense, makes it sound much more arbitrary than it is. Once you've defined the natural numbers and addition, you can't decide that 1+1=3; that 1+1=2 follows from the definition. The patterns you can find/create are limited by the objects/relations you start with. And that will be as true for us as any possible aliens, so they would create/find subsets of the same conceptual mathematical universe.
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Re: Does math represent truth?  
« Reply #19 on: Mar 23rd, 2012, 3:47pm »
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We are only looking at one particular but important aspect.  It seems there are a fair number of mathematical isms, very similar to those of politics. For example when I was examining the Dedekind cut, Grimbal drew my attention to another school, finitism.
 
As individuals we may start out with an open mind, but an institution might have to choose a main direction  for research.  
 
I am not a mathematician but I imagine that in the end you may also want to take side as an individual, since it may be impractical, or too much of a burden, to move forward on a broad front.
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Re: Does math represent truth?  
« Reply #20 on: Mar 26th, 2012, 5:33am »
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I'd say that the various schools of mathematical thought are more akin to religious than political leanings.
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Re: Does math represent truth?  
« Reply #21 on: Mar 26th, 2012, 8:39am »
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Hmm, I think there have been too few wars fought over mathematics for it to count as a religion.
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Re: Does math represent truth?  
« Reply #22 on: Apr 7th, 2012, 2:41pm »
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1 I agree as I mentioned before that natural numbers can be seen to have some kind of objective existence. I am more worried about the real numbers (in particular the uncountable indefinable real numbers, UIRN.  If you accept these you might also accept that all true but unprovable statements can be elevated to the status of axioms, thereby rendering the natural number system complete in Godel’s sense. This process is no more spooky that the formation of the UIRN, and the amount of added axioms is not likely to be higher than uncountable.)
 
2 Mathematics, in the astrological form was very important both in decisions regarding wars and peace some time ago. In the changes in Egypt when Hypatia was killed, her mathematics probably constituted  part of a potentially competing spiritual guidance which needed to be destroyed.
 
3 Today there seem to a negative relation between mathematics in general and wars (not a specific kind of mathematics however, and not relating to specific axioms).  Today’s anti-scientific beliefs, discussed elsewhere on this site, seem to go hand in hand with an aggressive foreign policy.  Also the de-coupling  of cause and effect (e.g. weapons of mass destruction visavis the invasion of Iraq) is an insult to mathematics since a cause and effect relationship do not exist in real life but only in models thereof – most often with mathematical components.  
 
4 What I really want to say, however, has to do with what we may imagine a person from some 10 millennia in the future thinks. The person would say that mathematics was used in its astrological form and a millennium later it was used to facilitate technical progress, later perhaps to be followed by other kinds of uses. Such uses might include very large scale experiments perhaps on philosophical or esthetical grounds, quite possibly related to mathematical axioms. People may take divided views on such issues, in large scale (I would hope friendly) discussions. (If I had access to advanced sociological terminology, I would probably sound even more convincing.)    
 
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Re: Does math represent truth?  
« Reply #23 on: Apr 8th, 2012, 1:49am »
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on Apr 7th, 2012, 2:41pm, Mickey1 wrote:
If you accept these you might also accept that all true but unprovable statements can be elevated to the status of axioms, thereby rendering the natural number system complete in Godel’s sense.
All undecidable statements can be used to extend a logic system; however, this always gives you two choices since the opposite statement will also be undecidable. Therefore you can't simply add all undecidable theorems to a logic and keep it consistent. Which statements are undecidable changes as soon as you add another axiom to a logic.
Anyway, Godel's incompleteness theorems conclusively proves you can't make a logic (that contains arithmetic) complete by adding axioms without making it inconsistent. From the arithmetic embedded in the logic it you can prove that there is an unprovable statement in the logic; and adding axioms can't restrict you to proving less in a logic system, so this won't change.  
Also, the natural numbers are a model, not a logic. So it simply has certain properties; you can add axioms to a logic that describes a given model, but you won't know if it describes the model correctly if the logic is incomplete. Adding undecidable theorems as axioms does not change that, so it still describes the model incompletely, since you don't know and can't know if the additional axioms describe any property of the model.
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Re: Does math represent truth?  
« Reply #24 on: Apr 13th, 2012, 4:39am »
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I have a modest and narrow goal  mentioning these real numbers. I am primarily contemplating the spooky use of real numbers which cannot be defined and therefore cannot be referred to, and I am looking for a similarity for true statements about natural numbers.
 
Regarding the undecidable statements, I think it makes more sense to assume they are true, since that is what they are. I can even prove this by using Godel’s theorem. The problem, I assume, is not their validity but the reference process:  
 
In my "proof" (using Godel) I am referencing undefinable -  and therefore unreferencable - statements. We cannot examine their validity, since we cannot know what they are. Perhaps the set of these statements is infinite, perhaps even uncountable. That would establish a parallel between that set (of statements) to the set of the undescribable real numbers.
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