The problem of Self-Referential systems
- Thread starter Write4U
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exchemist
Valued Senior Member
Gödel's Incompleteness Theorem is not exactly news. It's been around since the 1930s.
Write4U
Valued Senior Member
Yes, I was being dramatic. But not without some justification, IMO.
What I don't quite understand is the conclusion that "incompleteness" constitutes a fatal flaw? That sounds a little dramatic to me, given the spectacular success mathematics has enjoyed since it's discovery and the abstractness of the falsification. Mathematics cannot prove itself? Does it need to be self-referential in the abstract?
The self-reference is obvious in its functional reliability.
Besides, why should it be unusual that the Universe still has some hidden maths to teach us. Are we not a little arrogant to think we should know-it-all already?
exchemist
Valued Senior Member
If someone in the video (I haven't watched it) actually says it is a "fatal" flaw, that is a bit silly, to my mind.
Don't forget that there are other sources of bad weather.
The Omega Man
http://www.you.com.au/news/362.htm
He shattered mathematics with a single number. And that was just for starters, says Marcus Chown.
INTRO: Two plus two equals four: nobody would argue with that. Mathematicians can rigorously prove sums like this, and many other things besides. The language of maths allows them to provide neatly ordered ways to describe everything that happens in the world around us.
Or so they once thought. Gregory Chaitin, a mathematics researcher at IBM's T. J. Watson Research Center in Yorktown Heights, New York, has shown that mathematicians can't actually prove very much at all. Doing maths, he says, is just a process of discovery like every other branch of science: it's an experimental field where mathematicians stumble upon facts in the same way that zoologists might come across a new species of primate.
Mathematics has always been considered free of uncertainty and able to provide a pure foundation for other, messier fields of science. But maths is just as messy, Chaitin says: mathematicians are simply acting on intuition and experimenting with ideas, just like everyone else. Zoologists think there might be something new swinging from branch to branch in the unexplored forests of Madagascar, and mathematicians have hunches about which part of the mathematical landscape to explore. The subject is no more profound than that.
The reason for Chaitin's provocative statements is that he has found that the core of mathematics is riddled with holes. Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."
This is particularly bad news for physicists on a quest for a complete and concise description of the Universe. Maths is the language of physics, so Chaitin's discovery implies there can never be a reliable "theory of everything", neatly summarising all the basic features of reality in one set of equations. It's a bitter pill to swallow, but even Steven Weinberg, a Nobel prizewinning physicist and author of Dreams of a Final Theory, has swallowed it. "We will never be sure that our final theory is mathematically consistent," he admits.
Chaitin's mathematical curse is not an abstract theorem or an impenetrable equation: it is simply a number. This number, which Chaitin calls Omega, is real, just as pi is real. But Omega is infinitely long and utterly incalculable. Chaitin has found that Omega infects the whole of mathematics, placing fundamental limits on what we can know.
And Omega is just the beginning. There are even more disturbing numbers -- Chaitin calls them Super-Omegas -- that would defy calculation even if we ever managed to work Omega out. The Omega strain of incalculable numbers reveals that mathematics is not simply moth-eaten, it is mostly made of gaping holes... (MORE)
http://www.you.com.au/news/362.htm
He shattered mathematics with a single number. And that was just for starters, says Marcus Chown.
INTRO: Two plus two equals four: nobody would argue with that. Mathematicians can rigorously prove sums like this, and many other things besides. The language of maths allows them to provide neatly ordered ways to describe everything that happens in the world around us.
Or so they once thought. Gregory Chaitin, a mathematics researcher at IBM's T. J. Watson Research Center in Yorktown Heights, New York, has shown that mathematicians can't actually prove very much at all. Doing maths, he says, is just a process of discovery like every other branch of science: it's an experimental field where mathematicians stumble upon facts in the same way that zoologists might come across a new species of primate.
Mathematics has always been considered free of uncertainty and able to provide a pure foundation for other, messier fields of science. But maths is just as messy, Chaitin says: mathematicians are simply acting on intuition and experimenting with ideas, just like everyone else. Zoologists think there might be something new swinging from branch to branch in the unexplored forests of Madagascar, and mathematicians have hunches about which part of the mathematical landscape to explore. The subject is no more profound than that.
The reason for Chaitin's provocative statements is that he has found that the core of mathematics is riddled with holes. Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."
This is particularly bad news for physicists on a quest for a complete and concise description of the Universe. Maths is the language of physics, so Chaitin's discovery implies there can never be a reliable "theory of everything", neatly summarising all the basic features of reality in one set of equations. It's a bitter pill to swallow, but even Steven Weinberg, a Nobel prizewinning physicist and author of Dreams of a Final Theory, has swallowed it. "We will never be sure that our final theory is mathematically consistent," he admits.
Chaitin's mathematical curse is not an abstract theorem or an impenetrable equation: it is simply a number. This number, which Chaitin calls Omega, is real, just as pi is real. But Omega is infinitely long and utterly incalculable. Chaitin has found that Omega infects the whole of mathematics, placing fundamental limits on what we can know.
And Omega is just the beginning. There are even more disturbing numbers -- Chaitin calls them Super-Omegas -- that would defy calculation even if we ever managed to work Omega out. The Omega strain of incalculable numbers reveals that mathematics is not simply moth-eaten, it is mostly made of gaping holes... (MORE)
Write4U
Valued Senior Member
It's a good video. The narrator is does not judge but rather observes the current state of theoretical mathematics. He cites a lot of excellent information in support of and opposition to various aspects of abstract mathematics.
Write4U
Valued Senior Member
Perhaps, but the mathematics that apply to this small portion of the universe which we can observe and measure, have been spectacularly successful.
We have come up with equations for which we had no physical evidence, but is later discovered to apply to something completely unrelated than the original inquiry.
And IMO, that's where we may be misled. Our perspective may be too narrow due to our "local" spacetime coordinates and we have no direct experience with other dynamic parts of the universe.
But we are expanding our horizons and our understanding of the mathematics along with it.
Write4U
Valued Senior Member
"Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."
That's rather harsh. Seems to me that most of mathematics is founded on logic and then you'll have to explain why most of logic is untrue, or true by accident and if that proves Nature itself is not a mathematically ordered process.
To cite a few apparent theoretical contradictions seems to indicate the lack of sufficient human knowledge, rather than an inherent flaw in Nature's mathematics.
Most of math is based on axiom, testable and measurable natural phenomena. Moreover a lot of theoretical mathematics with no apparent specific utility is later proven to be useful in analyzing an unrelated phenomenon.
Most theoretical mathematicians such as astronomers claim that when they use some particular equation to test a theory, they find it is already used by nature and they freely admit that, rather than inventing natural mathematics, they are "discovering" pre-existing mathematics.
So for anyone to make grand claims that mathematics are flawed, he must realize that he is speaking of human mathematics, the symbolic translation of natural orderly processes . If there is a flaw, it's in the human part.
I believe the existence of the universe speaks for itself.
"input <--> function <--> output" = "garbage in, garbage out" . I submit the universe is not a garbage dump.
That's rather harsh. Seems to me that most of mathematics is founded on logic and then you'll have to explain why most of logic is untrue, or true by accident and if that proves Nature itself is not a mathematically ordered process.
To cite a few apparent theoretical contradictions seems to indicate the lack of sufficient human knowledge, rather than an inherent flaw in Nature's mathematics.
Most of math is based on axiom, testable and measurable natural phenomena. Moreover a lot of theoretical mathematics with no apparent specific utility is later proven to be useful in analyzing an unrelated phenomenon.
Most theoretical mathematicians such as astronomers claim that when they use some particular equation to test a theory, they find it is already used by nature and they freely admit that, rather than inventing natural mathematics, they are "discovering" pre-existing mathematics.
So for anyone to make grand claims that mathematics are flawed, he must realize that he is speaking of human mathematics, the symbolic translation of natural orderly processes . If there is a flaw, it's in the human part.
I believe the existence of the universe speaks for itself.
"input <--> function <--> output" = "garbage in, garbage out" . I submit the universe is not a garbage dump.
Sherlock Holmes
Registered Member
A bit of history might shed light on this question.
Back in the 1920s the established and respected mathematician David Hilbert proposed a research "program" to attempt to define the foundations of mathematics in such a way that certain paradoxes could be eliminated.
He wanted to define a single set of axioms from which all of mathematics could be deduced from, it was a grand idea.
Basically the program failed, the expected simple elegance was not possible, Bertrand Russel also admitted failure when his book Principia Mathematica also had to be abandoned.
So this is where the label "flaw" originates, it is in fact a flaw in human thought though, not in mathematics, the expectations set by Hilbert's Program was flawed.
exchemist
Valued Senior Member
Eh? Russell (and Whitehead) did not abandon Principia Mathematica. It was published, in 1910-13: https://en.wikipedia.org/wiki/Principia_Mathematica
Write4U
Valued Senior Member
Of course, theoretical mathematics can always be made to fail by proposing contradictory arguments. "you cannot fit a square peg in a round hole" does not pra logical esent a mathematical problem at all . It's an error in logic.
To try and prove mathematics with mathematics is the same as to try an prove time with time. Is there a flaw in Time?
The argument is that self-reference should always be a perfect equation, but that's just not true.
IMO, the Law of Necessity and Sufficiency does not prohibit a single reference.Whereas Necessity requires Sufficiency to be causal to change, Sufficiency does not require Necessity to be causal to change.
Natural Selection is a probabilistic
Sherlock Holmes
Registered Member
I meant to write that the goal was abandoned, the aspiration to derive all mathematics from a set of axioms was seen to be unachievable, because of what Gödel did.
Axioms and postulates are accepted on faith. They can't be proven for ALL cases.
Write4U
Valued Senior Member
Human maths cannot be proven for ALL cases. Nature has no such problem. The self-referential mathematics are an inherent potential of spacetime geography.
AFAIK, cause and effect is a deterministic equation, which suggests a mathematical function.
Write-4U;
Math is an abstract language used as a verification tool for science, in its various forms.
Just because it's useful in understanding the behavior of the physical world isn't sufficient reason to assume there is some form of mathematical influence in the universe.
Intangible relations, love, motivation, compassion, etc. can't be measured and fall outside of science.
Human knowledge has to rely on models that mimic or simulate the physical world.
The human mind is limited in its ability to form the basic concepts to understand how the universe works just as a pet dog is limited in understanding how television works.
To know how 'some things' work can't be extrapolated to 'all things'.
Your belief doesn't help our understanding.
Math is an abstract language used as a verification tool for science, in its various forms.
Just because it's useful in understanding the behavior of the physical world isn't sufficient reason to assume there is some form of mathematical influence in the universe.
Intangible relations, love, motivation, compassion, etc. can't be measured and fall outside of science.
Human knowledge has to rely on models that mimic or simulate the physical world.
The human mind is limited in its ability to form the basic concepts to understand how the universe works just as a pet dog is limited in understanding how television works.
To know how 'some things' work can't be extrapolated to 'all things'.
Your belief doesn't help our understanding.
Write4U
Valued Senior Member
You are speaking of human mathematics which are the symbolic representations of naturally occurring processing of values by means of logical (mathematical) functions.
Of course they can . All human emotions rest on electro-chemical processes in the body and brain, which are very much mathematically guided processes.
I agree, and you just confirmed the way the physical world works if it can be mathematically simulated with human symbolic maths.
Again I agree, but when theoretical mathematicians test a new theory against nature, they often find that nature has employed that mathematical equation all along and that we are not inventing math but discovering universal mathematics that existed there long before we recognized it.
Your words: "The human mind is limited in its ability to form the basic concepts to understand how the universe works just as a pet dog is limited in understanding how television work."
My position is that the universe has no such limitations.
I would suggest that perhaps your "unbelief" in universal mathematics limits your understanding.
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Write-4U;
A lottery can be described with math, but predictions are only probabilities, as with weather and quantum physics.
If you win the lottery it's just a coincidence.
A person wearing a good luck whatever will attribute their lack of danger to it, making associations where there are none.
People make associations whenever it's convenient or reassuring.
As for things like charity, a study will measure the degree in terms of monetary units.
How easy to be charitable if you are rich. What about motive?
If human thinking understood how the universe works, we could stop doing scientific research.
The reality is, they don't and can't know, and make it up as they go.
Truth will never be decided by an opinion poll!
A lottery can be described with math, but predictions are only probabilities, as with weather and quantum physics.
If you win the lottery it's just a coincidence.
A person wearing a good luck whatever will attribute their lack of danger to it, making associations where there are none.
People make associations whenever it's convenient or reassuring.
As for things like charity, a study will measure the degree in terms of monetary units.
How easy to be charitable if you are rich. What about motive?
If human thinking understood how the universe works, we could stop doing scientific research.
The reality is, they don't and can't know, and make it up as they go.
Truth will never be decided by an opinion poll!