Are there limits to knowledge in math (and science?)

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Are there limits to knowledge in math (and science?)

Post by Philosophy Explorer »

There are different levels of abstraction to math, one reason being is to generalize it to cover a wider area. It's natural to ask how far can one take this? If there are limitations, does this imply there's only so much we can know in math and science?

PhilX
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Re: Are there limits to knowledge in math (and science?)

Post by A_Seagull »

Philosophy Explorer wrote:There are different levels of abstraction to math, one reason being is to generalize it to cover a wider area. It's natural to ask how far can one take this? If there are limitations, does this imply there's only so much we can know in math and science?

PhilX
What do you mean by 'know' and 'knowledge' ?

Do you know anything at all? And if you do : How do you know it?

Maybe when you meaningful answers to those questions, it might be possible to move on to a discussion of their limits, if any.
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Re: Are there limits to knowledge in math (and science?)

Post by Treatid »

One of the limits on knowledge is that we cannot have absolute knowledge of any kind.

Mathematics proposes axioms as a starting point from which certain things can be known with respect to those axioms. In theory such knowledge is definite and unambiguous.

Unfortunately, axioms themselves cannot be defined or stated in a definite and unambiguous manner. Since logic and proof require definitely defined axioms... there is no definite knowledge as typically represented by proofs (i.e. there is no proof - of anything).

What is left is practical knowledge. We can build bridges and computers. We can fly to the moon and create a Global Positioning System.

Practical knowledge is the result of Trial and Error. We have discovered that some things work in certain situations. Mathematics represents part of this practical knowledge.

The limits of mathematical knowledge are the limit of physics (the physical world). If mathematics can be tested against experience then it represents knowledge.

If mathematics cannot be tested against real world experience then it has no meaning. There is no inherent meaning within mathematics... and mathematics is unable to generate inherent meaning from nothing.
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Re: Are there limits to knowledge in math (and science?)

Post by Reactor »

Philosophy Explorer wrote:There are different levels of abstraction to math, one reason being is to generalize it to cover a wider area. It's natural to ask how far can one take this? If there are limitations, does this imply there's only so much we can know in math and science?

PhilX
Abstraction to formula in mathematics can critically focus upon a particular subject or discipline to crystallize or simply show relationships that otherwise require many words.

While man continues to wonder and think about his real and imagined universe, mathematics and science will continue to evolve.

-- Updated October 10th, 2014, 1:51 pm to add the following --
Philosophy Explorer wrote:There are different levels of abstraction to math, one reason being is to generalize it to cover a wider area. It's natural to ask how far can one take this? If there are limitations, does this imply there's only so much we can know in math and science?

PhilX
There are no limitations.

Matter and energy and their effects upon each other already exist in the universe.

Our task is to recognize them and create explanations and mathematics in the attempt to understand and use them toward better survival.
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Re: Are there limits to knowledge in math (and science?)

Post by David_the_simple »

Kurt Godel proved that any axiomatic system powerful enough to represent itself would be consistent, but incomplete.

For example, before complex number were invented, the square root of -1 looked like a theorem, but was unknowable. The set of axioms was incomplete. We added imaginary numbers now to solve that problem, but sooner or later we will find a construct that looks like a theorem but is unknowable. Time to invent another axiom. Hence, by proxy of Kurt Godel's work, "there are no limits to knowledge in math"

As far as limits to knowledge in science, Jame Grier Miller's work in living systems introduced the concept of emergence. That phenomenon not expected out of the individual component would emerge from a system of components. If we entertain that constructs have no limit in complexity, then new phenomenon can be expected to emerge, hence again, "there is no limit to knowledge in science".
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Re: Are there limits to knowledge in math (and science?)

Post by Bohm2 »

I've never understood why this issue is often debated We are finite creatures with cognitive limitations like all other organisms. With respect to science, unless one believes we are Gods, then it's a given that there will be stuff beyond our cognitive reach/science and that we will never understand, regardless of how far science progresses. This does not necessarily imply that human knowledge cannot continue to progress/grow. To use an analogy used by those who espouse this view, the set of prime numbers is infinite but this does not exhaust the set of all natural numbers; that is, human knowledge can keep progressing and yet still have limits:
...so long as the class of accessible concepts is endogenously constrained, there will be thoughts that we are unequipped to think. And, so far, nobody has been able to devise an account of the ontogeny of concepts which does not imply such endogenous constraints. This conclusion may seem less unbearably depressing if one considers that it is one which we unhesitatingly accept for every other species. One would presumably not be impressed by a priori arguments intended to prove (e.g.) that the true science must be accessible to spiders. What is the relation between the class of humanly accessible theories and the class of true theories? It is possible that the intersection of these classes is quite small, that few true theories are accessible. There is no evolutionary argument to the contrary. Nor is there any reason to accept the traditional doctrine, as expressed by Descartes, that human reason is a “universal instrument which can serve for all contingencies.” Rather, it is a specific biological system, with its potentialities and associated limitations. It may turn out to have been a lucky accident that the intersection is not null. There is no particular reason to suppose that the science-forming capacities of humans or their mathematical abilities permit them to conceive of theories approximating the truth in every (or any) domain, or to gain insight into the laws of nature.
http://philos.nmsu.edu/files/2014/07/na ... ticism.pdf
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Re: Are there limits to knowledge in math (and science?)

Post by Philosophy Explorer »

Here's a specific math limitation (thanks to Galois):

He proved that in general, you can completely solve algebraic equations ranging from the first to the fourth degree, but beyond the fourth degree (with a few specific exceptions), you can NOT in general completely solve algebraic equations (but you can get estimates).

PhilX

-- Updated October 27th, 2014, 12:49 am to add the following --

Let me pose the question this way.

Is there any science field where there are any limitations to knowledge? (as for defining knowledge, there are many different types so I'll let the user specify which type he/she means).

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Re: Are there limits to knowledge in math (and science?)

Post by A Poster He or I »

Is there any science field where there are any limitations to knowledge?
The so-called hard sciences like physics do have limits to theoretical knowledge while at the same time they probably do not have any limits to practical knowledge.

What we KNOW, we know relative to a given epistemological framework. Scientific hypotheses and associated empirical data only become scientific KNOWLEDGE once they pass the testability requirements demanded by scientific epistemology.

Therefore, theoretical science does impart limits to what can be known scientifically because empirical validation is limited to spatio-temporal phenomena. In short, any hypothesis that posits non-spatiotemporal dynamics does not allow for empirical testability that meet the criteria of science. The hypothesis must either remain speculation or be accepted as knowledge outside of science's purview. Examples include M-theory and Bohmian Mechanics.

However, the speculative nature of untestable theory need not stop scientific progress in any practical way. The social sciences like psychology and anthropology readily concede that the interpretation of data is merely in accord with a given model, not universal and absolute, yet the fields have made progress. Physics can follow the same path. As example, the Industrial Revolution transformed society based only on knowledge of classical mechanics. Ignorance of quantum mechanics and Relativity theory did not impede its progress.
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Re: Are there limits to knowledge in math (and science?)

Post by Bohm2 »

A Poster He or I wrote: The hypothesis must either remain speculation or be accepted as knowledge outside of science's purview. Examples include M-theory and Bohmian Mechanics.
I noticed you picked Bohmian Mechanics. Since it's an interpretation of QM just like Copenhagen, MWI, Ensemble, TI, Objective collapse (GRW), etc. and since at present none make different predictions from Quantum theory, wouldn't you consider all of them speculative/philosophical?
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Re: Are there limits to knowledge in math (and science?)

Post by A Poster He or I »

Well, I actually feel that PhilX's question implicates Epistemology in its entirety, so my answer to your question is yes. But I doubt PhilX intended the scope to go so far, so I'll confine discussion to the matter of what qualifies as knowledge versus hypothesis in physics.

Most QM interpretations do not posit their own modifications to the standard QM formalism. The standard QM formalism is testable in accord with scientific epistemology and is therefore a symbolic representation of what we KNOW about QM. But Bohmian Mechanics adds its own additional formalism (viz., the Guidance equation) while at the same time it offers nothing to support the empirical correspondence of its variables to anything verifiable by science.

I am happy to concede that this makes no PRACTICAL difference at all because the results of QM experimentation have to be interpreted anyway, and interpretation occurs outside of what science can verify (in other words, we don't KNOW why only one value from the wave function's many possibilities is experienced upon observation). Since the Guidance Equation is consistent (though superfluous) with standard QM, it is just as good a way of interpreting QM as any other way, so long as one doesn't conclude that the terms of the Guidance Equation correspond to anything real.

So while Bohmian Mechanics does not constitute KNOWLEDGE about QM, it has practical potential for allowing us to see how non-locality might have a CAUSAL basis outside of normal spatiotemporal limitations, thereby spurring more speculation that could someday lead to practical results. It is an example of why I believe theoretical science, while self-limited, is potentially unlimited in its practical application. We may never know the ultimate WHY or HOW, but we may someday know everything about the WHAT.
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Re: Are there limits to knowledge in math (and science?)

Post by Apwbd »

Think of science and knowledge as a round spot of light in the middle of pure darkness. It's circumference is the border between knowledge and mystery. The bigger the spot gets, so does it's circumference. The more we discover, the more we realize that we don't know. So for now we can only assume that what we don't know is infinite, but there is no way to determine whether or not we will be able to know everything at some point in time. Therefore we can assume that there are limits to science at any point in time unless we know everything. Long story short, all the science fields are connected in some ways and since we don't know everything about any of them right now, math and every kind of science are ultimately limited.
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Re: Are there limits to knowledge in math (and science?)

Post by Bohm2 »

A Poster He or I wrote:Most QM interpretations do not posit their own modifications to the standard QM formalism. The standard QM formalism is testable in accord with scientific epistemology and is therefore a symbolic representation of what we KNOW about QM.
I don't follow this. What do you mean by the "standard QM formalism" and "symbolic representation of what we KNOW about QM"? ? Are you arguing that the informational and/or instrumental approach isn't another interpretation like the others? If you are, consider this quote by Fuchs who is one of the leading figures from the epistemic/informational camp:
To take a stand against the milieu, Asher had the idea that we should title our article, “Quantum Theory Needs No ‘Interpretation’.” The point we wanted to make was that the structure of quantum theory pretty much carries its interpretation on its shirtsleeve—there is no choice really, at least not in broad outline. The title was a bit of a play on something Rudolf Peierls once said, and which Asher liked very much: “The Copenhagen interpretation is quantum mechanics!” Did that article create some controversy! Asher, in his mischievousness, certainly understood that few would read past the title, yet most would become incensed with what we said nonetheless. And I, in my naivet´e, was surprised at how many times I had to explain, “Of course, the whole article is about an interpretation! Our interpretation!”...The question is completely backward. It acts as if there is this thing called quantum mechanics, displayed and available for everyone to see as they walk by it—kind of like a lump of something on a sidewalk.

The job of interpretation is to find the right spray to cover up any offending smells. The usual game of interpretation is that an interpretation is always something you add to the preexisting, universally recognized quantum theory. What has been lost sight of is that physics as a subject of thought is a dynamic interplay between storytelling and equation writing. Neither one stands alone, not even at the end of the day. But which has the more fatherly role? If you ask me, it’s the storytelling. Bryce DeWitt once said, “We use mathematics in physics so that we won’t have to think.” In those cases when we need to think, we have to go back to the plot of the story and ask whether each proposed twist and turn really fits into it. An interpretation is powerful if it gives guidance, and I would say the very best interpretation is the one whose story is so powerful it gives rise to the mathematical formalism itself (the part where nonthinking can take over). The “interpretation” should come first; the mathematics (i.e., the pre-existing, universally recognized thing everyone thought they were talking about before an interpretation) should be secondary.
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http://arxiv.org/pdf/1207.2141v1.pdf
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Re: Are there limits to knowledge in math (and science?)

Post by A Poster He or I »

I don't follow this. What do you mean by the "standard QM formalism"...
The standard QM formalism is constituted by the accepted mathematics of quantum mechanics. Traditionally it can take several forms such as wave mechanics, matrix mechanics or Dirac's transformational formalism. Later developments such as quantum electrodynamics have also become standard. All of these different formalisms can be shown to be equivalent to each other. By contrast, the Guidance Equation of Bohmian Mechanics is not accepted as standard formalism for QM.
...and "symbolic representation of what we KNOW about QM"? ?
These formalisms have survived empirical testing in accordance with scientific methodology. Indeed, quantum electrodynamics is the most accurate scientific theory ever invented in terms of correspondence to empirical results. The mathematical formalism is the "language" of the theory. It now constitutes knowledge, not hypothesis (according to the epistemology of science); what we know about the dynamics of quanta, as opposed to being merely speculative.
Are you arguing that the informational and/or instrumental approach isn't another interpretation like the others?
No. If anything, I'm suggesting that it is Bohmian Mechanics whose followers see it as more than just another interpretation, specifically those followers who would presume that the terms of the Guidance Equation correspond to objective elements in reality.
If you are, consider this quote by Fuchs who is one of the leading figures from the epistemic/informational camp:
The quote makes valid points that I agree with wholeheartedly. Those points are one reason why I am more-or-less a Copenhagenist: The Copenhagen Interpretation seems to be the least befuddled about where the data ends and the storytelling begins. CI recognizes that the very instruments that yield data are part of the story too (i.e., the interpretation).
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Re: Are there limits to knowledge in math (and science?)

Post by Bohm2 »

A Poster He or I wrote:No. If anything, I'm suggesting that it is Bohmian Mechanics whose followers see it as more than just another interpretation..
That can be said of all followers of any interpretation, I think.
A Poster He or I wrote:Those points are one reason why I am more-or-less a Copenhagenist: The Copenhagen Interpretation seems to be the least befuddled about where the data ends and the storytelling begins. CI recognizes that the very instruments that yield data are part of the story too (i.e., the interpretation).
I'm sure you know the criticisms against CI, but the same can be said of all interpretations. I favour DeBroglie's double solution model because of the experimental and theoretical work done by Couder and Bush with macroscopic quantum-like systems:

http://math.mit.edu/~bush/?page_id=484 http://dotwave.org/tag/couder/

-- Updated October 29th, 2014, 6:12 pm to add the following --
Bohm2 wrote:
A Poster He or I wrote:No. If anything, I'm suggesting that it is Bohmian Mechanics whose followers see it as more than just another interpretation..
That can be said of all followers of any interpretation, I think.
A Poster He or I wrote:Those points are one reason why I am more-or-less a Copenhagenist: The Copenhagen Interpretation seems to be the least befuddled about where the data ends and the storytelling begins. CI recognizes that the very instruments that yield data are part of the story too (i.e., the interpretation).
I'm sure you know the criticisms against CI, but the same can be said of all interpretations. I favour DeBroglie's double solution model because of the experimental and theoretical work done by Couder and Bush with macroscopic quantum-like systems:

http://math.mit.edu/~bush/?page_id=484

http://dotwave.org/tag/couder/
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Re: Are there limits to knowledge in math (and science?)

Post by Marsh8472 »

There probably are human limitations to learning. Maybe we could make AI that can discover more for us and surpass those limitations.
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