Tut, tut – how comically predictable. There you go again, vastly overreaching what was said or implied.
Never mind that your own stance is succinctly encapsulated by “Well hey, it must be a solid construction because Mechanist says so, right?” And never mind “Semper necessitas probandi incumbit qui agit.” Your unwittingly hypocritical impostures really are inordinately hilarious at times.
But here’s something I can assert with considerable confidence: If matters were anywhere near as simple as your puerile pretensions intimate, we’d not be having these exchanges. That would be because lots of people have been working on the problem for a very long time and who are a great deal smarter than you and I combined. So do carry on, please. The amusement value just keeps mounting.
True, it is kind of hard to “underreach” or even comment when a cute equation is given that does not actually add anything. My mistake there, apologies.
Aren’t you “vastly overreaching what was said or implied”? Mmmm…
I am interested in people’s opinions on the matter. These are obviously not all possible options or solid descriptions of reality and its possibilities. Feel free to contribute in a constructive manner though, it can be done.
If the following three are proposed:
Physical: Entities composed of matter and/or energy and their related properties that can be measured without changing their identity. We observe (mentally) that physical entities exist.
Mental: Mental events are related to intellectual processes of thought, perception, memory, emotion, will, imagination and intentionality. We observe the mental actions of other physical agents.
Platonic mathematical universals: This can be described as mathematical entities that exist independently of mental events. They are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. They are impossible to construct in the physical world, and hard to imagine in the mental world (Mintaka).
Why would anyone want to think 2) and 3) even exist?
Some people think “that 2) and 3) don’t actually exist where 2) is just fluctuations caused by 1). Nothing more than wind blowing.”
I would think that the reality of mathematical truths is self evident, and exists independently of wether we think about them or not. This is massively supported by observation. How many examples can you think of in 5 minutes? I’ll get you started with one:
A solar eclipse takes place a few thousand years ago. It caught all and sundry completely by surprise, because at the time nobody had the mathematics to predict such and event. So the local king got fired, a few oxen were sacrificed etc, and the astrologers/-onomers of the day record the date of the event after the fact. Today, years later, even a 10 year old kid can use one of several free amateur astronomy programs too verify that this eclipse should have taken place, just like it did. The point is that things happen in a highly predictable manner, and it happens wether we bothered to predict them or not. The predictability becomes revealed once one grasps the laws that govern them.
What remains a fascinating mystery though, is why the physical world should appear to “obey” the mathematical world in this way. Thank goodness it does, because we can work out how long a car trip would take at a given speed. And we can buy summer clothes in advance at discount prices.
I think the order of influence is as follows:
The Mathematical world controls the Physical world controls the Mental world.
Presumably, that would be the same perfect Platonic mathematical world that gave us the annoyingly self-referential and insurmountable Gödel Incompletenesses.
I vaguely remember this from the book [i]Fermats Last Theorem [/i] by Simon Singh. It boils down to the futility of trying to prove certain mathematical truths that contains too many logical steps, not so?
ETA: From Mefiante’s Wiki reference:
The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory.
If such an unprovable truth exists, why not just call it axiomatic and be done with it? Why should the impossibility of proof subtract anything from what is mathematically true and what isn’t?
Er, not really. It’s an in-principle limitation that exists on any axiomatic formal system of a certain minimum complexity. Stated in a loose but heuristic way, it says that there are things that cannot formally be known or demonstrated to be either true or false. For example, the parallel postulate of Euclid is a formally undecidable proposition in the general scheme of geometry: taking it as true leads to ordinary plane geometry, whereas taking it as false yields several, equally valid non-Euclidean geometries. Without the latter, it would not be possible to put General Relativity on a sound mathematical foundation. Another example is the so-called “continuum hypothesis” that characterises the relationship between countable and uncountable infinities (e.g. integers vs. real numbers). Taking it as either true or as false, each premise produces valid, though different, algebras. These in turn may one day have profound implications for gauge field theories in subatomic physics.
First, it’s not just unprovable truths but unprovable falsities too, and there is no rigorous way that the one can be formally distinguished from the other. Second, it’s not just one such truth (or falsity) per formal system; it’s an infinity of them. Third, adding an unprovable truth as an axiom to a formal system just produces another formal system of greater scope that will have its own infinitude of undecidable propositions, so you’re not much better off. Fourth, if you are diligent about it, you’d end up with two formal systems, one in which P is axiomatic, and the other in which not P is. Each of these can yield its own result for a given problem. Which one do you choose?
Because, as said, Gödel Incompleteness speaks not just to what’s true, but also to what’s false. If it were only relevant to unprovable truths, it would be trivial (and self-contradictory) because it would imply that any undecidable proposition must be true, and hence de facto decidable. I think the problem is that the Wikipedia article suggests – perhaps too strongly – that Gödel Incompleteness only entails certain things being mathematically true but unprovably so. Instead, it should rather be viewed as a bifurcation point at which a formal system can, if suitably diddled, give rise to two or more conflicting but equally valid answers. Postmodernists will no doubt be thrilled at this prospect but for the exact sciences this can be a bit of a bummer.
Thanks for your patient and chrystal clear reply 'Luthon. Yes, I did miss this point. And it does of course change everything. Ambiguity in the world of numbers. I must admit this rattles my cage a bit. But what about all those perfect predictions that it makes about the observable world - surely that is at least evidence that something accurate and consistent is in charge of physics? Is the Platonic Mathematical world blown out of the water?
Concur.
Mintaka Anti Gödel activist and disillusioned skeptic
Thanks for the compliments, guys! An interested audience makes it worthwhile. :-*
I’m hardly Jesus, but if I were, that’d be a rhetorical question… ;D
I have done a little lecturing and tutoring in the past but now one of my roles is that of an informal mentor.
Bear in mind that Gödel’s work is quite recent and that most if not all areas of mathematical study can trace their roots back to some or other practical problem, even if it is by a lengthy chain of removes. In that view, it could be argued that the mathematical world depends on what we observe as reality. At present, the implications of Gödel’s work are relevant predominantly to the foundations of mathematics, rather than to the usefulness of extant mathematics. However, they will inevitably determine the direction of new mathematical studies: If some mathematical construct cannot be firmly grounded, its use in descriptions of reality will have some looming question marks hanging over it. For example, in QM there is a technique called “renormalisation” (basically, the cancelling of infinite terms against one another, a strict mathematical no-no) that seems to work splendidly but which has yet to receive rigorous validation.
Also, mathematics is in many ways an endeavour to abstract and to generalise into ever-expanding domains in order to find common patterns among the objects of its study. It could be then that the Platonic universe is foam-like and vastly greater than one might currently suppose, and that we have merely had a peek into one tiny bubble of it, namely that one which bears most directly and immediately on our perceivable reality. But I am speculating here. The “unreasonable effectiveness of mathematics” (as one observer has put it) remains a baffling mystery. And it is hardly clear if reality (physics) is subordinate to some prior mathematical realm or if mathematics is just the most convenient possible medium for apprehending certain aspects of reality. Both of these schools of thought have their respective proponents. Personally, I think it’s more a case of parallelism than dependence in that we construct idealisations of real-world problems and by manipulating them in certain ways, we are able to abstract from them elements that are perhaps not obvious or that find application in other areas, indicating some deeper physical connection. The majority of our most elegant and useful mathematical models of reality arise from novel and inventive recombinations of existing mathematical ideas; though there are some exceptions, it is rather rare that such a model requires some wholly new mathematics.
I seriously doubt it. However, the point is that even a supposedly perfect, ideal, non-material world comes with some surprising rules and caveats, and, by extension, that its relationship to reality is anything but simple and unambiguous.
agreed on the clear and understandable explanations(s). Thanks for that - especially as this discussion mostly goes over my head anyway. despite that, allow me a silly question: are you guys implying that even if something can be mathematically proven (and as a result we would assume that it is true) that, if you are a clever mathematician, it could also mathematically be disproved? and if that is the case, why bother trying to prove anything? or are the clever maths-fundies working on figuring out which proof/refutement is more proof and hence more true than another?
The first thing to note here is that one must distinguish carefully between “truth” and “validity.” A mathematical proof may be valid and yet not true. This can happen if one of the assumptions (or axioms) is not true but the rest of the proof is logically impeccable. As an example, it is true (and provably true) that the sum of the three interior angles of a triangle is 180° if one assumes planar triangles. It is not true, e.g., in the case of spherical geometry (the sum will be greater than 180°).
If a given proposition can be proved both true and false in the same formal system then that particular system is inconsistent because anything can be proved in it. It is thus also complete because its proofs encompass all statements, be they true or not. Such a pathological system is obviously not very useful, but such a “can be proved either way” arrangement is not at all what is implied by Gödel incompleteness. Instead, Gödel incompleteness says that a system cannot be simultaneously consistent and complete. This means that there will be propositions that can be framed within a consistent formal system that cannot be proved to be either true or false (but not that they can be proved both ways). Propositions of this kind are called formally undecidable because there is no formulaic way of deciding whether they are true or false from within the formal system in which they arise using its axioms and inductive and deductive rules of inference.
Such propositions, though often very hard to identify, can be the wellspring of new mathematics because with them one can construct new formal systems, one in which such a proposition is axiomatised as true, and one or more others where it is false. Remember here that a proposition can be true in only one way, but false in several ways. To illustrate: If I say “Pi is three” then that statement can be true if and only if Pi is indeed three; the statement is false if Pi is any value other than three, and there are infinitely many such non-three value to choose from. The point here is that undecidable propositions open the door on new mathematical vistas because it allows mathematicians to investigate and develop what-if scenarios: “What if this proposition is taken as true?” “What if it is false in way X?” “And in way Y?” Broadly speaking, such questions characterise what happens at the cutting edge of mathematics.
It’s hard to see how there can be degrees of truth in mathematics, i.e. how one statement can somehow be closer to a mathematical truth than another. In mathematics, a proposition is either axiomatic, provably true or provably false, or formally undecidable. There’s no spectrum from true to false, just those extremes, and any proof or disproof is either valid (and thus fully convincing) or it contains errors.
;D once again an incredibly well put together response. thanks. i will never be a mathematician, but i think i understand the gist of where my original confusion started at least. wow it really is true what they said back in school, isn’t it - math people are really smart.
