I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really. — Zuhair
It all breaks down as limits are approached: — John Gill
That's ironic, the numbers approach infinity (limitless), as the condition approach the limit — Metaphysician Undercover
This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences. — Metaphysician Undercover
Not so for Planck Time. You'll need a real, live physicist to discuss this properly. It used to be that this limit was variable according to some physical features. — John Gill
I don't see where you differ with me. Mathematics can also speak of patterns that had not been yet observed! Because it tackle all possible structures in an unlimited manner. That's what I meant when I said *before-hand*, if we had good mathematics about ellipses, parabola, hyperbola, etc.., even before we observed the movements of planets, that knowledge would make it easier for the astronomer to discover the pattern of movement of those planets, because as I said many times humans don't see what they don't look for. — Zuhair
f you have the descriptive arsenal before-hand, you'll predict easily the behavior of matters with fewer observations because it would look familiar to what you have experienced in say the platonic world about those orbits. — Zuhair
Notice that we didn't coin the mathematical structures describing orbits (ellipses, hyperbola,etc..) after the observation had been made, we actually imagined it form more trivial observations on our planet, then we freely contemplated more variety of structures in the platonic imaginary world, this free contemplation is what made us arrive at those orbit mathematical structures way before any application was discovered. — Zuhair
And I think that's one of the most important jobs of mathematics, to supply such descriptive arsenal that objects in our world can possibly follow. I'd say perhaps, the particle physics objects move along some paths that we don't have the descriptive arsenal necessary to match them with, that's why we remain in ignorance about them. — Zuhair
So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems. — Metaphysician Undercover
Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others. — Zuhair
Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements. — Zuhair
Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallelity postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair
Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair
So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we known in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it should be observed if we are to have mathematics help enrich our knowledge about our world. — Zuhair
I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility. — Metaphysician Undercover
Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem — Metaphysician Undercover
Quick question: did you-all settle on a definition (at least for your present purposes) of "reality?" Yes or no is fine. If you did I'll go find it. If you did't....the reality of Platonic objects. — Metaphysician Undercover
I'm confused by the distinction actual vs potential infinity?
From wikipedia I get:
Potential infinity is a never ending process - adding 1 to successive numbers. Each addition yields a finite quantity but the process never ends.
Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity.
In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {.
Can someone explain this to me? Thanks. — TheMadFool
Is kind of why I asked about a working definition. It seems to me all those matters depend on context and what is required in and for the context. Kind of a shame you-all didn't. A good topic deserves good grounding. — tim wood
Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that nn is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition. — quickly
Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.
Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception. — Metaphysician Undercover
This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics. — quickly
You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language. — Metaphysician Undercover
Well one answer (the wrong answer) is we can go on dividing forever by 2 (say) so there must be an actual infinity of reals in the interval.
But we can only go on dividing forever in our minds - if we tried this in reality, we'd never finish dividing (process goes on forever - we'd never finish) - so the possibility of infinite division is just a figment of our imagination (like its possible to levitate in your imagination - but not in reality). — Devans99
From my perspective, ZFC has unsound axioms concerning the nature of objects, as we discussed earlier. Therefore any proof using ZFC is unsound. — Metaphysician Undercover
“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.” — Uncle Bertie
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