wtf

Do you mean in full detail? No, because it would have to include itself and every particle in the universe. Not enough energy in the universe to do that with. But if you mean could we have a really good model of the laws of physics, we already have those. They can simulate the first few seconds of the big bang and the subsequent evolution of the universe. This is old hat.

 

Why do you think AI has anything to do with this? It's humans who build the models. It's important not to get new age-y about all this stuff. Strong AI has been a complete failure and has produced nothing since the idea gained currency in the 1960's. Weak AI of course plays chess and drives cars. Impressive but very specialized problem domains. And whose achievement is it? The computer's? Or the armies of designers and mathematicians and programmers who build the clever little gizmos? The first thing to know about AI is how to separate out the breathless hype from the reality.

Sorry for the delay but we have been without power until now. The best I can do for now is to re-iterate my earlier post in a slightly different form

Sorry about your power loss but the recent pace is fine for me. It might have been me who pulled the plug

 

 

I am aware that, on forums such as this it is considered a hanging offence to disagree with the sacred Wiki, so let us say I have confused you. Specifically FORGET the term "transition function".

I'm just grasping at straws to follow your posts. FWIW here is a screen shot from Introduction to Differential Geometry by Robbin and Salamon. This is from page 59 of this pdf. https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf

 

 

They use the term transition map exactly as I've used it. But no matter, we can call them something else. But it's clear what they are, you are in agreement even if you prefer to use a different name.

 

 

We have 3 quite different mappings in operation here. The first is our homeomorphism: given some open set [math]U \subsetneq M[/math] that [math]\varphi:U \to R^n[/math]. Being a homeomorphism it is by definition invertible.

 

Suppose there exist 2 such open sets, say [math]U_\alpha,\,\,U_\beta[/math] with [math]U_\alpha \cap U_\beta \ne \O[/math]. In fact suppose the point [math]m \in U_\alpha \cap U_\beta[/math], so that [math]\varphi_\alpha:U_\alpha \to V \subseteq R^n[/math] and [math]\varphi_\beta:U_\beta \to W \subseteq R^n[/math].

 

So the composite function [math]\varphi_\beta \circ \varphi_\alpha^{-1} \equiv \tau_{\alpha,\beta}:V \to W \in R^n[/math]. One calls this an "induced mapping" (but no, [math]\varphi_\alpha^{-1}[/math] is not a pullback, it's a simple inverse)

 

Your Wiki calls this a transition, I do not. So let's forget the term.

Ok. I agree with all your notation so far. As I say it took me the duration of your power outage for all this to become clear so feel free to pretend the power's out as I work to absorb subsequent posts.

 

But note that single points in [math]V,\,\,W[/math] are Real n-tuples, say [math](\alpha^1,\alpha^2,....\alpha^n)[/math] and [math](\beta^1,\beta^2,....,\beta^n)[/math], so that image of [math]\tau_{\alpha,\beta}((\alpha^1,\alpha^2,....,\alpha^n))= (\beta^1,\beta^2,....\beta^n)[/math]

Yes, entirely clear.

 

So the second mapping I defined as: for the point [math]m \in U_\alpha [/math], say, the image under [math]\varphi_\alpha[/math] is the n-tuple [math](\alpha^1,\alpha^2,....,\alpha^n)[/math] likewise for [math]\varphi_\beta(U_\beta)= (\beta^1,\beta^2,....,\beta^n)[/math] then there always exist projections [math]\pi_\alpha^1(\alpha^1,\alpha^2,....,\alpha^2)= \alpha^1[/math] and so on, likewise for the images under [math]\pi_\beta^j[/math] of the n-tuple [math](\beta^1,\beta^2,....,\beta^n)[/math].

Perfectly clear.

 

Note that since the [math]\alpha^j[/math], say, are Real numbers this is a mapping [math]R^n \to R[/math].

 

So the composite mapping (function) [math]\pi_\alpha^j \circ \varphi_\alpha \equiv x^j[/math] is a Real-valued mapping (function) [math]U_\alpha \to R[/math] and the n images under this mapping of [math]m \in U_\alpha[/math] is simply the set [math]\{\alpha^1,\alpha^2,....,\alpha^n\}[/math] and the images under this mapping of [math]m \in U_\beta[/math] is the set [math]\{\beta^1,\beta^2,....,\beta^n\}[/math] so that [math]x^j(m) = \alpha^j[/math] and [math]x'^k(m) = \beta^k[/math]

Yes.

 

The [math]x^j,\,x'^k[/math] are coordinate functions, or simply coordinates

Ok so we are identifying the coordinates with the projection mappings composed on the charts that produce them.

 

The coordinate transformations I referred to are simply mappings from [math]\{x^1,x^2,....,x^2\} \to \{x'^1,x'^2,....,x'^n\}[/math], they map (sets of) coordinates (functions) to (sets of) coordinates (functions) if and only if they refer to the same point in the intersection of 2 open sets. This mapping is multivariate - that is, it is NOT simply the case that say [math]f^1(x^1)=x'^1[/math] rather [math]f^1(x^1,x^2,....,x^2)=x'^1[/math].

Yes this is clear to me.

 

Note that the argument of [math]f^j[/math] is a set, not a tuple, appearances to the contrary

I take this to mean that [math]\{f^j\}_{i=1}^n[/math] is a set of maps where [math]f^j = \pi_j \varphi_\beta \varphi_\alpha^{-1}[/math], is that right?

 

I hope this helps.

Yes very much.

 

It also seems I may have confused you slightly with my index notation - but first see if the above clarifies anything at all.

Yes much better. Of course the couple of days I spent working through this in my own mind helped a lot too.

 

P.S I am generally very careful with my notation.

Maybe I should leave that remark alone Let me just say that I sometimes find it productive to work through points of murkiness in your exposition. I'm ready for the next step and do feel free to take this as slowly as you like. Also if you have any particular text you find helpful feel free to recommend it. There are so many different books out there.

I think I understand what you're saying. In my notation, you are using [math]\beta^i[/math] as both the value of the [math]i[/math]-th coordinate of the [math]\beta[/math]-representation of some point [math]m \in U_\alpha \cap U_\beta[/math]; and also as the function [math]\pi_i \varphi_\beta \varphi_\alpha^{-1}[/math] that maps the [math]\alpha[/math]-representation of some point [math]m[/math] to the [math]i[/math]-th coordinate of the [math]\beta[/math]-representation of [math]m[/math].

 

That's how I'm understanding this. You're taking the [math]i[/math]-th coordinate to be both the function and the specific value for a given [math]m[/math]. It's a little bit subtle. The REAL NUMBER [math]\beta^i[/math] changes as a function of [math]m[/math]; but the FUNCTION [math]\beta^i[/math] does not.

 

Is that right? I want to make sure I'm nailing down this formalism.

 

Secondly I believe that you are a little confusing or inaccurate when you say the transfer maps (without the extra projection at the end) go from [math]U[/math] to [math]U'[/math]. Rather the transition maps go from [math]\varphi_\alpha(U_\alpha \cap U_\beta)[/math] to [math]\varphi_\beta(U_\alpha \cap U_\beta)[/math] and back.

 

Since the charts are homeomorphisms so are the transfer maps in both directions. And I've read ahead on Wiki and a couple of DiffGeo texts I've found, and I see that if the transfer maps are differentiable or smooth then we call the manifold differentiable or smooth. That makes sense. We already know how to do calculus on Euclidean space.

 

So I'm a litle confused again ... the charts themselves don't have to be differentiable or smooth as long as the transfer maps (on the restricted domain) are. Is that correct? So for example the charts could have corners outside the areas of overlap? Perhaps you can help me understand that point.

[math](a - b)^2 = (b - a)^2[/math]. Your question is like noting that [math](-5)^2 = 25[/math], then asking if we are given [math]25[/math], how do we know if we "started" with [math]5[/math] or [math]-5[/math]. And the answer is that we don't. Squaring loses information. The squaring function maps two different values to the same value so you can't reliably go backwards.

 

ps -- There is a philosophical aspect to this point. If we view an equation as a statement that two different-looking expressions point to the same object, then from [math]a = b[/math] we may infer [math]b = a[/math]. Equality is a symmetric relation.

 

However if we regard an equation similarly to a formula in chemistry, a statement that one thing yield another thing via some process; then from [math]a = b[/math] we may not necessarily infer that [math]b = a[/math]. Not all transformations may be reversed.

 

This impacts our daily lives in the form of Internet security. Public key cryptography is based on the fact that [math]3 \times 5 = 15[/math] is a computationally easy problem; while [math]15 = 3 \times 5[/math] is not.

 

In your example we have a transformation (squaring) that's not reversible at all, because it loses information.

 

In the multiplication/factoring example we have a transformation that is in principle reversible, but one direction is computationally more expensive than the other.

 

When you say two things are equal, you have to be careful what you mean.

I'm replying to your post #27 which said ...

 

Let's move on to the really interesting stuff ...

I commented on the first half earlier. Now to the rest of it.

 

First there's a big picture, which is that if we have a manifold [math]M[/math] and a point [math]m \in M[/math], then we may have two (or more) open sets [math]U, U' \subset M[/math] with [math]m \in U \cap U'[/math]. So [math]m[/math] has two different coordinate representations, and we can go up one and down the other to map the coordinate representations to each other.

 

My notation in what follows is based on this excellent Wiki article, which I've found enlightening.

 

https://en.wikipedia.org/wiki/Atlas_(topology)#Transition_maps

 

The notation is based on this picture.

 

 

We have two open sets [math]U_\alpha, U_\beta \subset M[/math] with corresponding coordinate maps [math]\varphi_\alpha : U_\alpha \rightarrow \mathbb R^n[/math] and [math]\varphi_\beta : U_\beta \rightarrow \mathbb R^n[/math]. I prefer the alpha/beta notation so I'll work with that.

 

Also, as I understand it the coordinate maps in general are called charts; and the collection of all the charts for all the open sets in the manifold is called an atlas.

 

If [math]m \in U_\alpha \cap U_\beta[/math] then we have two distinct coordinate representations for [math]m[/math], and we can define a transition map [math]\tau_{\alpha, \beta} : \mathbb R^n \rightarrow \mathbb R^n[/math] by starting with the coordinate representation of [math]m[/math] with respect to [math]U_\alpha[/math], pulling back (is that the correct use of the term?) along [math]\varphi_\alpha^{-1}[/math], then pushing forward (again, is this the correct usage or do pullbacks and pushforwards refer to something else?) along [math]\varphi_\beta[/math].

 

So we define [math]\tau_{\alpha, \beta} = \varphi_\beta \varphi^{-1}_\alpha [/math]. Likewise we define the transition map going the other way, [math]\tau_{\beta, \alpha} = \varphi_\alpha \varphi^{-1}_\beta [/math].

 

I found it helpful to work through this before tackling your notation.

 

 

So suppose the coordinates (functions) in [math]U[/math] are [math]\{x^1,x^2,....,x^n\}[/math] and those in [math]U'[/math] are [math]\{x'^1,x'^2,....,x'^n\}[/math] and since these are equally valid coordinates for our point, we must assume functional relation between these 2 sets of coordinates.

Now I feel equipped to understand this.

 

We have [math]m \in U_\alpha \cap U_\beta[/math]. Then I can write

 

[math]\varphi_\alpha(m) = (\alpha^i)[/math] and [math]\varphi_\beta(m) = (\beta^i)[/math], with the index in both cases is the [math]n[/math] in [math]\mathbb R^n[/math]. I don't think we talked about the fact that the dimension is the same all over but that seems to be part of the nature of manifolds.

 

Question: You notated your ordered n-tuple with set braces rather than tuple-parens. Is this an oversight or a feature? I can't tell. I'll assume you meant parens to indicate an ordered [math]n[/math]-tuple.

 

Also you referred to the coordinates as functions, and you did that earlier as well. I'm a little unclear on what you mean. Certainly for example [math]\alpha_i = \pi_i \varphi_\alpha(m)[/math], in other words the [math]i[/math]-th coordinate with respect to [math]\varphi_\alpha[/math] is the [math]i[/math]-th projection map composed on [math]\varphi_\alpha[/math].

 

Are you identifying each coordinate with its respective projection map? That's perfectly sensible. You probably said that earlier.

 

 

 

For full generality I write

 

[math]f^1(x^1,x^2,....,x^n)= x'^1[/math]

[math]f^2(x^1,x^2,....,x^n) = x'^2[/math]

..............................

[math]f^n(x^1,x^2,....,x^n)= x'^n[/math]

Aha. This took me a while to sort out. What is [math]f^i[/math]? Putting all this in my notation, we have

 

[math]f^i(\alpha^1, \alpha^2, \dots, \alpha^n) = \beta^i[/math].

 

So we seem to be starting with the [math]\alpha[/math]-coordinates of [math]m[/math], using the transfer map [math]\tau_{\alpha,\beta}[/math] to get to the corresponding [math]\beta[/math]-coordinates; then taking the [math]i[/math]-th coordinate via the [math]i[/math]-th projection map.

 

Therefore we must have [math]f^i = \pi_i \tau_{\alpha,\beta} = \pi_i \varphi_\beta \varphi_\alpha^{-1}[/math].

 

As far as I can tell this is the equation that relates your notation to mine. Have I got this right?

 

 

 

 

Or compactly [math]f^j(x^k)=x'^j[/math].

I undersand that. But note that it's ambiguous. Does [math]f^j[/math] act on the real number [math]x^k[/math]? No, actually it acts on the [math]n[/math]-tuple [math](x^k)_{k=1}^n[/math]. So if we are pedants (and that's a good thing to be when we are first learning a subject!) it is proper to write [math]f^j((x^k)_{k=1}^n)[/math]. Whenever we see [math]f^j(x^k)[/math] we have to remember that we are feeding an [math]n[/math]-tuple into [math]f^j[/math], and not a real number.

 

But since the numerical value of each [math]x'^j[/math] is completely determined by the [math]f^j[/math], it is customary to write rhis as [math]x'^j= x'^j(x^k)[/math], as ugly as it seems at fist sight*.

This is very interesting. Let me say this back to you. [math]m[/math] has [math]\beta[/math]-coordinates [math](\beta^i)[/math]. And now what I think you are saying is that we are going to identify the coordinate [math]\beta^i[/math] with the map [math]f^i = \pi_i \varphi_\beta \varphi_\alpha^{-1}[/math]. Is that right? We identify each [math]\beta[/math]-coordinate with the process that led us to it! Very self-referential

 

This is what I understand you to be saying, please confirm.

 

ADDENDUM: No I no longer understand this. [math]f^i[/math] doesn't play favorites with some particular [math]\beta^i[/math]. It makes sense to say that [math]f^i[/math] maps [math]\varphi_\alpha(m)[/math] to the [math]i[/math]-th coordinate of [math]\varphi_\beta(m)[/math]. But it's a different [math]f^i[/math] for each [math]m[/math].

 

I think I am confused. I should sort this out before I post but I'll just throw this out there.

 

This is the coordinate transformation [math]U \to U'[/math]. And assuming an inverse, we will have quite simply [math]x^k=x^k(x'^h)[/math] for [math]U' \to U[/math]

Ok I had to think about this. Two points.

 

* Each [math]f^i[/math] is a map from [math]\mathbb R^n[/math] to the reals. It inputs an [math]n[/math]-tuple that is the [math]\alpha[/math]-representation of a point [math]m[/math]; and outputs a single real number, the [math]i[/math]-th coordinate of the [math]\beta[/math]-representation of [math]m[/math].

 

So the only way to make sense of what you wrote is to that the the collection of all the [math]f^i[/math] 's are the coordinate transformations.

 

Actually what I understood from the Wiki article is that the transfer maps were the coordinate transformations. So maybe I'm confused on this point. Can you clarify?

 

* There's actually a little swindle going on with [math]\varphi_\alpha[/math]. At first it was a map from [math]U[/math] to some open subset of [math]\mathbb R^n[/math]. But in order to pull back along [math]\varphi_\alpha^{-1}[/math] we have to restrict the domain to the image [math]\varphi_\alpha(U_\alpha \cap U_\beta)[/math]. So we don't really have a map from [math]U[/math] to [math]U'[/math] in your notation; but only from their intersection to itself.

 

Can you clarify?

 

Notice I have been careful up to this point to talk in the most general terms (with the 2 exceptions above). Later I will restrict my comments to a particular class of manifolds

It doesn't seem to matter at this point what the topological conditions are. It's all I can do to chase the symbols.

 

 

* Ugly it may be, but it simplifies notation in the calculus.

I think I'm with you so far. Just the questions as indicated. Two key questions:

 

* How the transition maps can be said to be from [math]U[/math] to [math]U'[/math] when in fact they're only defined from the [math]\alpha[/math] and [math]\beta[/math] images, respectively, of the intersection. I'm just a little puzzled on this.

 

* Your notation [math]x'^j= x'^j(x^k)[/math]. First I thought I understood it and now I've convinced myself [math]x'^j[/math] depends on [math]m[/math].

 

* And now that I think about it, the transition maps are from Euclidean space to itself, they're not defined on the manifold.

 

I'm more confused now than when I started working all this out.

Are you talking about the transition maps? I'm working through that now. The Wiki page is helpful. https://en.wikipedia.org/wiki/Manifold

 

ps ... Quibbles aside I'm perfectly willing to stipulate that the topological spaces aren't too weird. Wiki says they should be second countable and Hausdorff. Second countable simply means there's a countable base. For example in the reals with the usual topology, every open set is a union of intervals with rational centers and radii. There are only countably many of those so the reals are second countable.

 

Interestingly Wiki allows manifolds to be disconnected. I don't think it makes a huge difference at the moment. I can imagine that the two branches of the graph of 1/x are a reasonable disconnected manifold.

So you don't like my use of the term "transitive".

You are using the term in a highly nonstandard way and your exposition is unclear on that point.

 

 

Let's move on to the really interesting stuff, closer to the spirit of he OP (remember it?)

Very much so. I'm interested in why differential geometers and physicists are so interested in using dual spaces in tensor products when the algebraic definition says nothing about them. The current exposition of differential geometry is very interesting to me but not particularly relevant (yet) to tensor products.

 

The connectedness property mandates that, every [math]m \in M[/math] there exist at least 2 overlapping coordinate neighbourhoods containing [math]m[/math]. I write [math] m \in U \cap U'[/math].

I hope I may be permitted to post corrections to imprecise statements, in the spirit of trying to understand what you're saying. The indiscreet topology is connected but each point is in exactly one open set. Perhaps you need the Hausdorff property. Again not being picky for the sake of being picky, but for my own understanding. And frankly to be of assistance with your exposition. If you're murky you're murky, I gotta call it out because others will be confused too.

 

I'm still digesting the rest of your post.

I must say that this use of the term Hausdorff is quite different from what I've learned about the term. In my understanding, asking if that property is transitive is meaningless.

 

A topological space is Hausdorff if it separates points by open sets. That is, given any two points [math]x, y[/math], there are open sets [math]U_x, U_y[/math] with [math]x \in U_x[/math], [math]y \in U_y[/math], and [math]U_x \cap U_y = \emptyset[/math].

 

For example the real numbers with the usual topology are Hausdorff; the reals with the discrete topology are Hausdorff; and the reals with the indiscreet topology are not Hausdorff.

 

I confess I have no idea what it means for the Hausdorff property to be transitive. It's not a binary relation. It's a predicate on topological spaces. Given a topological space, it's either Hausdorff or not. It would be like asking if the property of being a prime number is transitive. It's a meaningless to ask the question because being prime is a predicate (true or false about any individual) and not a binary relation.

 

Given a pair of points, they are either separated by open sets or not. Of course for each pair of points you have to find a new pair of open sets, which is what I think you are saying.

 

Historical note. Felix Hausdorff was German mathematician in the first half of the twentieth century. In 1942 he and his family were ordered by Hitler to report to a camp. Rather than comply, Hausdorff and his wife and sister-in-law committed suicide. https://en.wikipedia.org/wiki/Felix_Hausdorff

ps -- Let me just say all this back and, being a pedantic type, clarify a couple of fuzzy locutions.

 

The indivisible pairing [math]S,T[/math] is called a topological space. Note that [math]T[/math] is not uniquely defined - there are many different subsets that can be found for the powerset.

Minor expositional murkitude. I'd say this as: For a given set [math]S[/math], various topologies can be put on it. For example if [math]T = \mathcal P(S)[/math] then every set is open. That's the discrete topology. The discrete topology is nice because every function on it to any space whatsoever is continuous. Or suppose [math]T = \{\emptyset, S\}[/math]. This is called the indiscrete topology. No sets are open except the empty set and the entire space. And the everyday example is the real numbers with the open sets being countable unions of open intervals. [This is usually given as a theorem after the open sets have been defined as sets made up entirely of interior points. But this is a more visual and intuitive characterization of open sets in the reals].

 

Now often one doesn't care too much which particular topology id used for any particular set, and one simply says "X is a topological space". I shall do that here.

This is actually interesting to me. Do they use unsual topologies in differential geometry? I thought they generally consider the usual types of open sets. Now I'm trying to think about this. Hopefully this will become more clear. I guess I think of manifolds as basically Euclean spaces twisted around in various ways. Spheres and torii. But not weird spaces like they consider in general topology.

 

Ouch, this already over-long

Well if anything it's too short, since this is elementary material (defined as whatever I understand ) and I'm looking forward to getting to the good stuff. But I hope we're not going to have to go through the chain rule and implicit function theorem and all the other machinery of multivariable calculus, which I understand is generally the first thing you have to slog through in this subject.

 

If you can find a way to get to tensors without all that stuff it would be great. Or should I be going back and learning all the multivariable I managed to sleep through when I was supposed to be learning it? I can take a partial derivative ok but I'm pretty weak on multivariable integration, Stokes' theorem and all that.

 

And really finally, one defines projections on each n-tuple [math]\pi_j:(u^1,u^2,....,u^n)\to u^j[/math], a Real number.

 

So the composite function is defined to be [math]\pi_j \circ h = x^j:U \to \mathbb{R}[/math]

 

Elements in the set [math]\{x^k\}[/math] are called the coordinates of [math]m[/math]. They are functions, by construction

What you are doing with this symbology is simply putting a coordinate system on the manifold. We started out with some general topological space, and now we can coordinatize regions of it with familiar old [math]\mathbb R^n[/math]. All seems simple conceptually. In fact my understanding is that "A coordinate system can flow across a homeomorphism."

 

I know these things are called charts, but where my knowledge ends is how you deal with the overlaps. If [math]U, U' \subset S[/math], what happens if the [math]h[/math]'s don't agree?

 

Ok well if you have the patience, this is pretty much what I know about this. Then at the other end, I do almost grok the universal construction of the tensor product and I am working through calculating it for multilinear forms on the reals. This is by the way a very special case compared to the algebraic viewpoint of looking at modules over a ring. In the latter case you don't even have a basis, let alone a nice finite one. So anything involving finite-dimensional vector spaces can't be too hard

I wasn't aware of this, so pointing that out only made me feel stupid.

I'm terribly sorry if my exposition had that effect. We're all hopelessly ignorant of so many things.

 

 

 

It made me reflect on myself and realize I shouldn't go in-depth talking about things I don't understand. There is a lot I don't know and one cannot even know what he doesn't know. I would like to know a lot more but there is no shortcut to this. Everyone who knows a lot about something has spent a lot of time getting to that position. So I learned some humility.

I should have mentioned earlier that the point I'm making, that there are not necessarily any laws of nature, is a minority opinion. I'm pretty sure the average working physicist thinks they are discovering the laws of nature, not just inventing prettier lies. My opinion is the extremist alternative one here.

 

You could then ask ''so why did you assume that there are absolute and correct laws of physics?''. I wasn't aware that there was a second option to begin with. I now realize that this is a philosophical issue, rather than simply a logical one.

Ok glad I made my point, even at the expense of some confusion along the way.

 

When the ancients looked at the night sky they saw hunters and bulls and crabs and all the other constellations. Humans see patterns even when there are no patterns. It's certainly fair to say that the constellations in the sky are artifacts of our minds and not anything that's actually there. Orion the hunter is something we made up and has nothing to do with the universe.

 

But now when we see patterns in the data from an atom smasher, how do we know we're not doing the same thing? Perhaps our physics is nothing more than imaginary patterns in our mind and nothing to do with the actual universe as it is.

 

That's my point. But if you asked a physicist, they'd almost certainly say that they trying to discover the actual laws of nature. They'd regard my point as profoundly wrong.

 

In fact my understanding is that the average physicist would not regard their work as worth doing if they couldn't think of themselves as trying to discover the true laws of the universe. I think they're wrong about that. But they haven't asked me, actually.

 

I thought we agreed on coins so I did not know how to expand upon discussing them. I understood Strange's comment and it's in line with I was thinking. You do realize I understand we can give odds only if we don't know the result, but knowing the result dismisses any odds?

I think that's really the answer to the question. Sometimes probability refers to an inherent quality in the thing we're observing; but usually it's more to do with the state of our ignorance.

 

 

 

Yes, I believe that's what I was getting at. I never thought of it as a philosophical question. I didn't think it could be.

It goes back to the old argument about free will versus determinism. Maybe everything that happens in the world, including the fact that I'm writing this post, was determined at the moment of the big bang.

 

Or maybe I have free will. In which case, why do I have free will to choose which words to write, but no free will to decide to fly into the air by flapping my arms?

 

As you asked, why is my body bound by the law of gravity, but my thoughts aren't? Isn't my brain a physical thing?

 

These are good questions and they are matters of philosophy. The philosophers are pretty sharp. It's trendy these days for scientists to mock philosophy, but I think it's the scientists who are actually wrong about the true nature of their enterprise. They're makding useful models, not necessarily discovering ultimate truth.

 

 

Sure. But I'm not sure what about coins we could discuss anymore so that it was in the spirit of this thread. I think we reached a simple conclusion.

If a philosophically-oriented discussion on an Internet forum actually reached a satisfactory conclusion, that is a remarkable achievement

 

Nah. I +1'd you for the honest answers, even if they were harsh. Science does not care for demeanor, only truth.

No, I was crabby. I did and do apologize for that.

 

 

If you go back to my goblin example, that's exactly what I was saying. Sure, it's silly, but it conveys the point. I don't think it was fair to say I was talking about a god machine in THAT case. I have no objection for the other cases. The example was equivalent to what Strange said.

Fair enough.

So far so good at my end. I haven't forgotten this thread, I've been slowly working my way through the universal property of the tensor product applied to multilinear forms on the reals. I can now visualize the fact that [math]\mathbb R \otimes \dots \otimes \mathbb R = \mathbb R[/math], because you can pull out all the coefficients of the pure tensors so that the tensor product is the 1-dimensional vector space with basis [math]1 \otimes \dots \otimes 1[/math]. This is pretty simple stuff but I had to work at it a while before it became obvious.

 

I'm still curious to understand the significance of the duals in differential geometry and physics so feel free to keep writing, you'll have at least one attentive reader.

First of all, the issue of editing posts to add something of value is the possiblity of the other person not realizing that it was edited. I had seen you come to the thread and leave a couple of times and I wondered why you didn't respond. Only now I realize you actually did.

If the same person posts twice in a row, the forum software here merges both posts. I've gotten used to that so I wrote a short post in the morning and a longer one later. There's no way for anyone to know when a post's been substantially edited, and there's no way to write two posts in a row.

 

 

That being said, there is an enormous shift in attitude by you. I am quite surprised by it with this last post. Reading through it, I agree that my thoughts and posts were all over the place and most likely pointless, at least from this sub-forum's perspective.

However, your post made me feel bad, especially because I'm a person uneducated in mathematics seeking some clarification. You may not realize that some of the things you pointed out are not neccessarily obvious to someone illiterate in maths.

Please accept my apology. You're right, my last post was a little crabby. I should have spent more time to figure out how to dial it back.

 

I steered away again from the coin discussions only because you did. I had to respond to your last points regarding matters which do not concern the coin.

You're right. I'm interested in the philosophical aspects so I talked about them while saying we shouldn't talk about them. My bad again.

 

 

I am also insulted that you (as far as I've gathered) concluded that I'm taking this is in a spiritual direction or pushing some kind of ''god'' agenda. There is absolutely nothing spiritual about me. I am not a crackpot. My ignorance is exactly that, and any misunderstanding we may have between us stem from it.

Ok well here I'll stand by what I wrote. I'm using the phrase "God's machine" to summarize what you are saying. That there's a machine, but it doesn't actually obey the laws that physical machines must obey. It's programmed, but not with any physics that we know. It's programmed with the "true" physics of the world, which for all we know doesn't exist at all.

 

So this is a hypothetical magic box that you are using for your argument, and I'm just calling it the God machine. Because that's what it is. It works by no known physics, it can predict the future purfectly, etc. "God" is a perfectly fair characterization of such a device. Not in a religious sense, but in the sense that it is a hypothetical device that is omnipotent and omniscient. the two qualities most often associated with the religious God.

 

I can call it the Magic Box if you like. It's a hypothetical device that transcends all known physical law and can perfectly predict the future. It's a God box.

 

 

 

No. You're oversimplifying it. And once again, there is no ''god theory''. Lose the idea, please.

Am I misunderstnaing your hypothetical device? Works by no known laws of physics, has infinite storage and processing capacity, can perfectly predict the future? Isn't that what you are describing?

 

I am not providing or refuting any sort of god's physics law. I think you got irritated with me and chose to reduce my thoughts to stupid gibberish.

But you did say that. I made the point that even if you have a computer with infinite processing and storage capacity, if it's programmed with the currently known laws of physics, it will be inaccurate from day one. That's almost exactly what I said earlier.

 

And you replied:

 

Yes, I know. I was considering a hypothetical machine which would know the full and correct laws of physics and mathematics.

 

That's what you wrote. God's law. Not God of any particular religion, but the "true" and "ultimate" laws of nature. This is exactly what you wrote. "Full and correct laws of physics." What did you mean by that if I'm mis-characterizing it?

 

If I was crabby it was because by engaging your philosophical ideas I saw that I was encouraging them, when I'm trying to get you to separate out these metaphysical speculations ("... the full and correct laws of physics ...") and stick to simpler things that we can analyze. Coin flips for example.

 

But ok if I was crabby I apologize. But I'm probaby still coming off as crabby. You used the phrase "... the full and correct laws of physics ..." and it's a philosophical assumption to even think there is any such thing. And if these ultimate laws are unknown to us and may forever be unknown to us ... who knows them? God is the name I give to that knower of the ultimate answers. Again not in the religious sense, but what else can you call the knower of the unknowable?

 

Surely, by how I defined ''techical odds'' here, this is not a ''philosophical assumption''. This is a matter of definition, but essentialy true.

Please don't assume now I'm trying to introduce some new law of probability here.

I believe it is a philosophical assumption. The idea that there is any "true" probability of anything. Isn't that the question you're asking?

 

Also, saying that someone's thoughts are philosophical is almost synonimous with saying they are useless, or rubbish, at least in my view.

Oh no, I love philosophy and idle philosophical speculation. I'm just suggesting it should go in the philosophy section.

 

You've raised several philosophical questions. Are there "ultimate" laws of nature, and if so, can they ever be known to us? That's the philosophy of physics. Is the human mind subject to physical law? That question's as old as Descartes. It's philosophical.

 

I don't think philosophy is rubbish. I think it's important and interesting. It's just confusing the issue in this thread, because these are very complicated questions.

 

I'm not sure how to proceed from here.

Well you could accept my apology for being crabby earlier. But if you say that you have a magic machine that knows the ultimate laws of physics, I'm entitled to point out that those are philosophical assumptions. And it's not unfair of me at all to call that a God machine. It's a little like the God particle. It's just a name.

 

You were being rational and patient in posts prior to this one.

Well I don't know what to say. You want to talk about a machine that knows the ultimate laws of physics. And you are unhappy that I'm pointing out that you are making philosophical assumptions about the nature of the world. I don't know any other way to say it.

 

I suppose I may be misunderstanding you.

 

On my part I'm trying to get you to separate out the philosophy of the universe from the much simpler case of flipping a coin. I'm calling out your philosophical assumptions in order to get you to see that you are making philosophical assumptions. The idea that there is a "true" physics is a philosophical assumption, it's not a fact. I'm not saying philosophy is bad. I'm just trying to point out where you are making assumptions about the world that are not known or not proven or not even provable in theory.

 

And coming off crabby, I suppose. I'll plead guilty to that and throw myself on the mercy of the court.

 

I thought Strange really had the last word here with the example of flipping a coin and one person knows the result and the other person doesn't. One person's odds are 100% for whatever the flip is. The other person's odds are 50-50 because of their lack of knowledge. Anything we talk about beyond that is confusing the issue IMO. What do you think about that example?

Oh god. I wrote an elaborate and lenghty post addressing all your points which would, I think, make you understand all I'm saying.

Same thing happened to Fermat. He had a marvelous proof but lost it in a browser mishap.

 

What I usually do is hit Quote then copy/paste into a text editor, do my writing, then paste back into the browser. On those occasions when I forget to save my text file, my computer invariably crashes, losing my work. One is constantly fighting entropy.

 

====

 

Ok.

 

Yes, I know. I was considering a hypothetical machine which would know the full and correct laws of physics

What makes you think there are any such things?

 

What's the difference betweeen just saying God knows everything that will happen? Your example is equally mystic.

 

 

and mathematics.

The correct laws of math? Absolutely no such thing. Some geometries are Euclidean, some not. Some groups are Abelian, some not. Some set theories are well-founded (no sets can be members of themselves) others not.

 

Math is agnostic regarding truth. Logical consistency and interestingness are all that matter.

 

 

 

 

I don't even know if there is such a thing as knowing something 100%, but for example, it would have the solution to the n-body problem. So it wouldn't be programmed in accordance with human knowledge of physics, but it would rather have an absolute knowledge of physics, if such a thing is possible.

Thus, approximation error would hopefully be eliminated.

Ok fine, God knows everything. I see no reason to dance around your spiritulalism. A machine that doesn't work according to the known laws of machines, programmed by laws that we don't know even exist, that predicts the future.

 

Whatever. Why not abandon all that confusion and just say:

 

"Imagine God knows the future. What's the probability of a coin flip?"

 

Isn't that the sum total of what you're saying here?

 

OK, but only until there is a 1 in 1 chance we are understanding each other

After this last bit I think it's zero. My only point is that we should talk about the coins and stop talking about God machines. You only want to talk about God machines. I can't respond any more to those kinds of speculations.

 

Here is the distinction between technical and practical odds: Practical odds are the ones given due to inability to calculate. So, coin is a 1 in 2 odds and dice are a 1 in 6 odds simply due to our inability to calculate how they will land with a 100% accuracy.

Technical odds are the ones where the result is always calculated to a 100% certainty. Maybe odds is not the appropriate term here because the solution is known, hence no odds are neccessary.

Ok that's clear. Just so that it's also clear that the idea that there actually are true odds is a philosophical assumption. You have no evidence for it beyond a minor amount of local experience near our spacetime coordinates.

 

Just to clarify a bit further, consider the following hypothetical scenario:

 

There is a goblin sitting in a wooden box. This box has a square hole in one of the sides. The goblin has 10 pieces of paper, each containing a number from 1-10. He picks these on random and displays them through the square hole to me. Now, I have practical 1 in 10 (10%) odds of guessing which number he will display.

However, let's say his box has another hole on the side. If I take a peek through this hole, I can see which number he picks before he displays it through the hole. So, I have a 1 in 1 chance of guessing the number right. This is what I consider the technical odds; the ones I have enough information about to know the solution.

A goblin? Whatever. Just say you have a God machine and be done with it. But that assumption already incorporates several unspoken assumptions about the universe.

 

For the several-th and last time I say: Forget about the univers and split off a philosophy thread for that. It's irrelevant.

 

This is what I mean by reducing the odds to 1 in 1.

These may be confusing terms, but, again, although I would like to be educated in mathematics, I am not, so I have to make up my own terminology. I hope you understand what I mean by them.

I undertand you think that the universe is logical and ordered. There's no proof that's true, only local evidence. By local I mean what we've been able to see from earth in the past few thousand years. Very limited sample of observations.

 

 

Yes, but my hypothetical prediction machine would know these. It would always adjust its calculation by the shift in the gravitational constant. The machine doesn't have the lack of knowledge and misunderstandings that the humans do. But more on that later, since we are trying to simplify here.

Ok you have a God machine.

 

No, I am not claiming that in any way. I am simply noting that the laws of the universe are absolute and unchangeable, while human thought isn't.

That contradictory. Your thoughts are a product of your brain which is a physical thing. You can't have it both ways.

 

Why are you insisting on complicating a simple discussion of probability by now claiming that thought is not subject to the laws of physics but everything else is, and in fact not the human-discovered laws of physics but God's physics, for which not a shred of evience exists?

 

 

For example, a slightly abstract analogy:

 

If a human is regularly presented with a choice of 3 different ice-creams, he may alter his choices every time in a seemingly unpredictable pattern, while a machine might choose different ones, but in a predictable and mathematically presentable algorithm. Although the machine is way more advanced than a human in some regards, its choices are of the same predictability as the universe's choices regarding odds. Feel free to substitute ice cream with whatever is more apt.

So thoughts aren't physical? You are contradicting your God theory now.

 

To reiterate, if you give the machine the ability to flip coins, it would do so in an eventually predictable manner, and so the odds of its flips would be reucable to techical (1 in 1 odds), provided we have the ability to calculate the angle, speed, force etc. of its coin flips.

If you give a human a coin to flip, he wouldn't neccessarily do this in a predictable pattern.

This is too far afield for me to comment on anymore. You want to take this to the metaphysical speculation forum.

 

This is the closest you've come to understanding me. I suppose, in core, the first question could be used as synonimous to some questions I am asking here.

I used the example of thoughts to show that it is POINTLESS to drag the theory of mind into this simple discussion of philosophy.

 

You've doubled down by going on about the nature of the universe and the nature of mind, which does not according to you even obey God's laws of the universe. You no longer have a coherent line of argument at all.

 

 

Yes, it's not mathematics per se, nor is it strictly philosophy of the mind. That's why I said I don't know where I would place the thread. It does concern probability theory in the way that it asks the question IF human thought is based on probability.

You are all over the map and refuse to engage with the simple coin example.

 

I re-wrote the post hastily before I leave for work, so I'm sure something got left out. Luckily, this is a forum and we can always get back to other questions.

I will re-phrase my questions in a way that they are more mathematical and practical, instead of abstract and possibly philosophical. But first, I need to be sure we're understanding each other.

You've lost me totally. You only seem to want to engage in idle speculation about God's secret laws of the universe and how the mind doesn't obey them. That's not even philosophy, it's just late night dorm room chatter.

 

It's strange. When I write posts, they are logical and concise in my mind. They seem really simple. But a lot of the time, members say they do not understand what I'm talking about and it always results in back-and-forth complications like these. Maybe my thoughts are erratic, who knows. I can't tell from my perspective.

Writing clearly is hard. In my previous post I did my best to try to get you to focus down on the coin example, and you responded by spinning wildly into God's secret laws and how the mind doesn't obey them. This is not good communication IMO.

 

EDIT: I only call them technical ODDS because the whole term of ''odds'' is relative. In reality, there is nothing probable or improbable about a coin toss. It is always going to land the way it is supposed to land. Our ignorance and incapability to calculate is what makes the coin toss odds-based. I am aware of this.

In the same way, the term random is relative. Nothing is random when you possess enough data.

I'm afraid you've lost me entirely in this response. You haven't got a magic box containing God's law and if you did it wouldn't prove anything about anything.

 

I'm disappointed that what I wrote to you earlier was so unclear as to have prompted you to go off in these unproductive directions. I was trying to get you to focus on the coin, and forget the God machine and your contradictory ideas about how the the universe has absolute laws but the mind doesn't obey them. I feel that I failed totally to make my point earlier and only made things worse.

 

Forgetting for a moment the prediction aspects, there is a sense in which probability is (or can be) simply a reflection of our knowledge about the world.

 

Imagine you flip a coin and catch it without looking at it. Now you and your friend both know the odds of it being heads are 50:50.

 

But if you look at the coin, the probability for you changes to 0 or 1. But for your friend it is still 0.5.

This. You don't need a God machine to discuss this simple and clear idea.

No, your post sums my thoughts up well.

Good, then we are communicating.

 

 

I know the unpredictability is due to our ignorance. The only reason why we say coins have a 50/50 chance for landing on either side is because we do not posses the information to calculate how they will land with 100% accuracy.

Yes and no. All physical theories are approximate. Even if we have machines with infinite capacity and speed, yet if they must be programmed with algorithms that represent CURRENTLY KNOWN physics, they must already be in error from the start, and those errors will accumulate to produce incorrect results.

 

Newtonian gravity refined Aristotelian gravity ("All bodies move toward their natural place"); Einstein refined Newton. Some future genius will refine Einstein.

 

We never know the ultimate laws of the universe, only mathematical approximations. Nor do we have any way to know for sure that there are any "ultimate" laws at all. Maybe it's turtles all the way down.

 

So in a controlled experiment we can predict a coin flip with 99% accuracy or maybe even 99.9999999% accuracy but NEVER with 100% accuracy.

 

And if you are trying to predict the evolution of the universe, that uncertainty must necessarily introduce massive error.

 

We can predict coins but not universes. That's what my simplification is about.

 

I propose that we should discuss coins and not universes, at least for a while. Coin flips are a more clarifying case. We know coins, we don't know the universe.

 

 

 

That's why I made the distinction between practical odds and technical odds in the first post.

Ok. After reading your post some more I don't think I fully understand your definitions.

 

I consider practical odds to be the ones usually used by humans to decide the chance of something happening (1 in 2 for a coin toss, 1 in 6 for a die roll etc.). I consider technical odds to always be 1 in 1. I don't know if this is something that is taught in mathematics, but surely it must be correct and it is very simple to grasp.

No that can't be right. Do you mean that if there are two possibilities that the odds are equal? If I jump off a high place I might fly or I may flop. The odds aren't 50-50. Or are you saying that perhaps I'm only applying what I know about the world? In that case you're right. If I have no prior information at all, I guess everything's 50-50. But we're never that ignorant.

 

I may be missing your point here. What do you mean the "technical odds" are always 1-1? Maybe I'm misunderstanding your definition of technical odds. I thought that referred to the "real" odds based on the laws of the universe.

 

 

Maybe it's just a case of me asking an overcomplicated question to which no one knows the answer.

I think you're asking a simple question about coins and complicating it by making many unproven assumptions about the universe and our ability to model it and compute with those models.

 

The theoretical limitations of measurement and modeling are inherent even in the coin experiment. But with coins the measurement and approximation error is miniscule. When applied to the evolution of the universe, the uncertainty predominates and we can no longer predict.

 

That's why I say, let's just talk about coins. The good question you are asking is whether probability is inherent in the event itself, or if it's merely a measure of our ignorance.

 

In the context of coins, clearly the probability measures our ignorance, since coins are pretty much deterministic.

 

But as far as the universe, who knows. There's a laundry list of issues to be considered. Best not to talk about the universe. That's my opinion. Because to talk about the univers you have to work out your metaphysical beliefs about the universe, whether you think there are any actual laws at all, what the relation of those "real" laws is to our historically contingent theories of physics; and then after all that you still have to solve the problem of calcution error ... you'd simply never get to the end of this conversation and you'd never reach a conclusion.

 

 

 

Alright, really the simplest way I can re-phrase the question is:

 

All natural and physical processes have technical odds.

That's a philosophical assumption about how the world works. That the world logical and ordered, the way it appears to us. Just noting a hidden philosophical assumption in your worldview. And what are technical odds? I'm not clear on your precise meaning.

 

That is, they can be reduced to a 1 in 1 chance because they are ever-repeating and absolute. Gravity will always act the way it ever did. So gravity is predictable and reducable to a 1 in 1 chance.

My understanding is that this is in question, even among physicists. Perhaps the gravitational constant drifts over the years. It's certainly possible. The only observations we have are very nearby in spacetime.

 

Again, you are confusing the universe with our latest contemporary model of the universe. But if there is anything we know about science, it's that no theory is ever final; and that all theories are approximations.

 

The "real" law of gravity, if there even is such a thing, can not possibly be our current theory of gravity any more than it was Newton's or Aristotle's theory of gravity. Our current theory is our best approximation to what we observe. No more and no less.

 

 

 

Is the same true for humans?

 

Can human behaviour and choices be reduced to technical odds?

So ok, this is in my opinion another complication we should leave aside. Just for the moment. I just want to nail down the coins, develop a common understanding around that.

 

If the universe is one complication too far, what can we make of human nature and the human mind? Are our thoughts themselves just physical processes, subject to your deterministic prediction machine? Or are they ... what, exactly? If we are not physicalists, then what other choices are there?

 

Ok this is a deep and wonderful question. But it's not a question about probability theory. The nature of the universe, and the nature of the human mind, must be ruthlessly left aside. At least for the moment. Till we understand the coins.

 

I don't know if human thought is algorithmic, physical, subject to deterministic prediction. It's a great discussion, but one that clouds our conversation here.

 

Coins. Not universes. Not minds. Coins are simple. Universes and minds are way too hard.

 

 

Is it even technically possible to find an algorithm or pattern in their behaviour so you can predict their choices? Humans do not always act the same way as opposed to gravity, which does.

You should split off a thread in the philosophy of mind section.

 

But since I can't resist responding ... I will note that if you claim the human mind does NOT obey the laws of physics ... then what kind of mysticism or spiritualism do you believe in?

 

I find both physicalism AND anti-physicalism equally problematic. If we're ONLY machines, that's bad. And if we're not machines ... well what else is there? Damn good questions I say.

 

Thinking about it, it really may be a case of asking a question no one has the answer to.

Yes yes. But we CAN speak very sensibly about coins.

 

That's my suggestion. Abandon talk about universes and minds. Not because these aren't vital subjects. But because they're not subjects we can really tackle all at once.

 

But the coins, the coins are simple. Coin flips are pretty much deterministic, and the 1/2 probability measures our ignorance.

 

Still I only say "pretty much" deterministic. After all, we really don't know what causes things to happen in the world.

 

Well I apologize if I've let myself ramble but it's all interesting to me

Just for the moment, let me change the subject to simplify it. If my simplification is off base, let me know.

 

Suppose I have a perfect mechanical prediction machine. Whenever I am about to flip a coin, I input all known information about the angle and position of the coin on my thumb, the physical condition of my thumb, exactly how much strength is in my thumb today, etc.

 

When I do this, I can predict the result of the coin flip with perfect accuracy.

 

But one time, my prediction machine is in the shop. It's not available to me. So when I flip a coin, the odds are 50-50. Not because of the inherent unpredictability of the coin toss; but rather because of my own ignorance.

 

That's the core question about probability. Does it reflect something inherent? Or is it just a measure of our ignorance?

 

When it comes to coins, it's pretty clear that probability is only measuring our ignorance. Because they've done lab experiments where they very carefully control the input variables of the coin flip, and they can predict the flip with better than 50-50 results.

 

However when you bring in things lie evolution of the universe and human thought, you're introducing extraneous variables about which we know very little. My sense is that you're asking about whether coin flips are inherently unpredictable, or whether they're only unpredictable due to our ignorance.

 

Is this in line with your thoughts?

Thanks for the info.

I'm not sure how there can be a rounding error possible if you assume the machine posseses the exact values of every variable involved.

 

Ah, maybe I understand. Judging solely by the name of the ''n-body'' problem, you might be referring to the fact that some of these numbers will be infinite in nature (after the decimal point) and therefore, unusable in any kind of precise calculation.

 

But I will make the effort of reading and trying to understand the links before I proceed with the discussion.

Two different issues.

 

* You can't store the **EXACT** value of quantities in a finite calculating machine. (Unless you can. Maybe the universe is discrete and finite. Nobody knows).

 

* The n-body problem is a problem in physics. We can't solve the differential equations for the motion of even 3 bodies under mutual gravity, let alone n bodies.

 

But as I say if your argument can be done theoretically by imagining that we have a God machine, that's fine. Your logic can proceed from there.

 

The n-body problem goes back to the time of Newton and is still important today. The Wiki article is worth reading.

 

https://en.wikipedia.org/wiki/N-body_problem

 

ps -- To make this clear: I have no objection to your assuming a hypothetical prediction machine. Just be aware that such a thing is not physically possible as far as we know. Or at least as far as I know. I could be wrong.

 

pps -- I think in essence you are asking if probability is a measure of the true unknowability of things; or if it's only a measure of human ignorance.

 

For example they've done lab experiments where they show there is nothing at all random about coin flips, once you precisely control the force and angle and direction of the flip mechanically.

 

So I think you are asking a question about the philosophy of probability. Is that right?

Basically, to rephrase the question: there are certain algorithms and laws which govern how the universe works. These are objective and ALWAYS true. That's why they are laws. Humans don't work that way. They aren't neccessarily predictable. So I'm asking if you think there ARE algorithms and patterns on human decision regarding odds and if they can be acted upon. Would a sufficiently advanced machine be able to decode them?

Not by any conventional definition of machine.

 

On the other hand if you're asking if God can predict the future, then that's a question of theology.

 

The Wiki links on the n-body problem, chaos theory, and the stability of the solar system provide basic information on my point.

 

If by machine you mean something that can store a finite amount of information, then it's subject to rounding error, and these will add up over time making prediction impossible.

 

If by machine you mean something that can calculate with infinite amounts of information, that goes beyond the laws of physics and computer science. And EVEN THEN you can't solve the n-body problem.

 

So a machine given total knowledge of the present state of a deterministic system can't predict the future. That's my understanding.

 

But if you want to imagine a hypothetical future predicting machine called God, that's perfectly ok. Just realize you've gone beyond science into theology and fiction. Perfectly fine for a thought experiment, if it's helpful.

 

In fact in computer science they sometimes imagine devices called oracles that can solve unsolvable problems, and then they reason based on those. So if you want to imaging you have a magic prediction machine, that's perfectly ok. Just be aware that you're reasoning from a hypothetical that doesn't actually exist.

Now, if you were to task said-super advanced machine to calculate these odds after the big bang, it would give 1 in 1 odds, because it already possesses sufficient information to calculate this (it would be able to infer this from the physics and chemistry of the big bang etc.).

Of course, such a feat would require a retardedly strong machine, but the point still stands.

No, this turns out to be untrue. Even if you know the position and velocity of every particle of a deterministic system, you still can't predict the future. As an example, we still have no idea whether the solar system is stable under Newtonian gravity.

 

The main problems are:

 

* You can't solve the n-body problem.

 

* Rounding errors in the calculations will add up over time.

 

https://en.wikipedia.org/wiki/Three-body_problem

 

https://en.wikipedia.org/wiki/Chaos_theory

 

https://en.wikipedia.org/wiki/Stability_of_the_Solar_System

 

A highly recommended popular book on the subject is:

 

https://www.amazon.com/Newtons-Clock-Chaos-Solar-System/dp/0716727242

I am also suggesting that 0 times anyting is anyting.

You could invent a system like that but it wouldn't be a field, since 0 times anything is 0 in a field. The reason is that it's provable from the field axioms.

A x 0 = A

​0 x A = A

0 times anything is 0. That's provable from the field axioms. So the above can't be true in any field.

Studiot you must be making a point I don't understand to double down on this line of argument. I'm mystified to see you laboriously enumerate the field properties in the pursuit of demonstrating a manifestly false understanding of what a field is. Surely you don't think there's any difference between your ABCD field, your squares and triangles, and any other presentation of the unique field with four elements.

 

If someone asks you how many fields there are of order four, and you give any answer other than one, you are missing the entire point of how algebraic structures work.

 

We do not care about multiple representations of the same isomorphism class of objects. If a set of widgets satisfies the field axioms, then it has a 0 and a 1, no matter what you call them.

 

That's why I don't object to Conway telling me that 0 and 1 are objects that contain z1 and z2. It makes no difference. 0 and 1 are defined by their behavior, not their representation. Perhaps this point is not brought out clearly enough in elementary presentations. Isomorphism isn't a map between two different things. It's a statement that we have two different representations for the same thing. It's really no different than the distinction between 1/2 and 2/4. Two different expressions for the exact same thing. Perhaps you'll give this some thought.

 

To pick a simpler example, because to be fair the field of order 4 is a bit of a counterintuitive object, how many vector spaces are their of dimension two? I hope you'll agree that there is only one, even though it can appear in different guises.

 

In your latest picture, the square is 0 and the triangle is 1. Every field has a 0 and a 1. It's part of the definition.

 

I hope you will agree that there is nothing numeric about turning a square and a triangle into an axehead (or the inverse operation for those inverses that are specified in the tables).

That's an interesting remark. I see nothing about being "numeric," whatever that means, in the definition of a field. If you insist I could do the same trick with the real numbers (replacing each real number by one of uncountably many symbols) and then I could say that the real numbers aren't numeric either. That's just wordplay. The real numbers are the unique complete ordered field. Any representation will do.

z1 and z2 are NOT elements in a field. It is that they are elements of an element in a field.

Ok I can live with that. What is the purpose of characterizing 1 and 0 as being sets that contain z1 and z2? What happens next?

I would also point out that the proposed axiom has no relevance to addition or subtraction.

Well if z1 and z2 are elements in the field, then you are required to say what is their sum and product with each element of the field including themselves and each other. That's because a field is closed under addition and multiplication. In other words if x and y are any two elements of a field, not necessarily distinct, then you have to define x + y and xy. If you have elements that can't be added or multiplied then you might have some algebraic structure, but it wouldn't be a field.

 

https://en.wikipedia.org/wiki/Field_(mathematics)

 

I don't know if this would be helpful to you but for example suppose we define two formal symbols x and y, and we want to consider the collection of all finite sums of elements ax + by where a and b are real numbers. Then the structure we get would be the vector space of dimension two, in other words the usual Cartesian plane.

 

Perhaps this is the kind of structure you have in mind. You have your formal gadgets z1 and z2, whatever they are, and you want to be able to multiply them by scalars. So you want to look at vector spaces or their generalization modules over a ring. https://en.wikipedia.org/wiki/Module_(mathematics)

 

But if you don't tell me what z1 times z2 is, then you haven't got a field, by definition.

I deliberately chose a field, that contains neither zero nor one.

Every field contains 0 and 1 and they are distinct. A = 0 and B = 1 in your example, which can be read directly off your plus and times tables. As a check you see that B + B = A, or 1 + 1 = 0. That's because the field of order 4 has characteristic 2.

 

Surely you understand that changing the names of the elements of an algebraic structure makes no difference at all.

Can you give examples for familiar fields such as the rationals, the reals, and the integers mod 5?