I don't see any algebra, here...
I was speaking more generically, in that when a nay sayer makes his case its in terms of things which are pretty basic, they never phrase their arguments in terms of more advanced mathematics or physics. This suggests that they are either a) unaware there's more to relativity than just doing coordinate transformations, b) that they've only just come across relativity and so haven't advanced that far or c) they don't understand more advanced material, though often they'll claim the opposite.
For instance, Jack_ throws around words like 'decidable' yet can't use the word 'proof' properly and all his algebra is 1st year stuff (at best). He claims to understand the posts I make in regards to bundles yet he never uses the much more powerful formalism of bundles to make his case. Why make life harder for yourself if you have the tools at your disposal to strengthen and streamline your argument? Unless, of course, he doesn't understand.
If the observer on the train were to have two synchronized clocks at his midpoint position, and then place one clock on each end of the train, and return to his midpoint position, when he looks at the clocks, they will not read the same thing.
This is false. As already explained to you such a result is in direct manifest contradiction to one of the postulates of SR and thus SR doesn't predict that the clocks will be different. And its a good job it does because that's in agreement with what experiments say!
That phenomena is due to the fact that the observer has a velocity.
Velocity with respect to whom? The ground? A plane in the sky? Jupiter? Motion is
relative. If the train is on a perfectly smooth track so that it doesn't jiggle about when moving
relative to the ground then alone straight horizontal lines of track the person on the train is in an inertial frame. He
cannot work out his motion relative to the ground without looking out the window. His claim "I am in an inertial frame" is true and allows him to consider himself at rest. With the clocks being at rest
with respect to him then their times will read the same from his point of view.
Consider the case where all the windows are blacked out so the person on the train cannot get any information outside of the train's interior. By your logic he can work out his velocity by comparing clock times, that the amount the clock times disagree by allows you to find his speed. But this is not possible as the fact he's in an inertial frame means he's 'blind' to the motion of things outside of the train. He can't tell if the train is stationary or moving relative to the Earth, provided that motion is constant. If a specific speed could be obtained then it would be associated to a preffered frame and even if that were the case there's absolutely no reason to think it would be the ground's rest frame (particularly given the Earth is not in inertial motion, it rotates).
There is no way the train observer will see the synchronized clocks as reading the same from his midpoint position, when the clocks are placed at each end of the train.
You're wrong, as he's in an inertial frame. If you put two sync'd clocks at opposite corners of the room you're in right now and then sat in the middle you'd see them reading the same times, right? Right. After all you and they are stationary, the ground isn't moving underneath you, right? Wrong. They read the same times because you are stationary
relative to them and in that frame you're at the midpoint between them. The motion of the Earth under your feet (or not) is irrelevant. After all, why pick Earth? What about the Moon, you're not stationary relative to the Moon, should the clocks tell you the relative motion between you and the Moon, as you claim it does between the train passenger and the Earth? Its the same setup, 2 clocks, an observer and some container within which they set which is moving relative to something else.
Unless you believe the universe has a prefered frame and that frame is some point of ground on the Earth's surface your argument falls apart, independent of whether or not it passes experimental tests.
Light travels independently of the train, at c. The time it takes the light to travel to the train observer's midpoint position is different for each strike. It is true that the train observer remains at the midpoint of the train, but when opposite ends of the train have light sources, and the lights are simultaneously activated (easily done with two lights and a common switch at the midpoint, activated by the observer), during the time it takes for light to travel the half length of the train, the observer also travels away from one light, and towards the other. That creates a situation that the light from one source has less distance to travel than the light from the other source, until it impacts the observer.
You're describing the system from the point of view o an observer at rest with respect to the ground and made the mistake of thinking that the person on the train sees exactly the same sequence of events in the same time frames, which isn't the case. This is the counter-intuitive nature of special relativity, you've gotten stuck on the first big hurdle I'm afraid.
Can we stick to Einstein's example?
Sure, but I'd request you actually take the time to read his writings on the matter and then to learn the basics of special relativity because presently I'd say you lack the mathematical tools and conceptual understandings required to get your head around the counter-intuitive nature of the non-Euclidean geometry at play in relativity.
As I said previously, look at threads started by the poster Jack_ (if you aren't he) and you'll find this setup has been beaten to death, full Lorentz transform examinations have been done, including diagrams, all in an effort to explain to Jack what we're now going to end up explaining to you.
Are you saying that the light from the back of the train to the observer travels the same distance as the light from the front of the train to the observer?
You really do need to spend some time finding out what SR is about, how its constructed and how it describes such systems, as you're currently displaying considerable naivety about it. If you don't know what SR actually says and why it says it it's a little unwise to try to argue that its wrong in what it says, particularly when those people you're trying to discuss SR with actually know some SR.
That is simply impossible
Given that comment we're now moving away from somewhat uninformed naivety to wilful ignorance and arrogance. You have simply utterly failed to understand relativity, either conceptually and quantitatively and given the pseudo-rhetorical question I quoted you saying just above suggests you haven't really made any effort to understand. Am I wrong in this evaluation, have you put in a fair chunk of time in order to try to understand SR? If not then its both silly and arrogant to say "Its obviously wrong, its impossible!" when you know full well you've not got a perfect understanding of the subject matter.
That is an indisputable fact for any observer in this universe.
And you know this how? Have you done your own experiments to test this premise? If so, what was it and when did you do it? If not then explain why you think you're justified in making what seems to be an unjustified claim. Those people who have done experiments in relation to relativity have found relativity to be vindicated, the issue of simultaneity and motion altering clocks is fundamental in the design of the GPS network, as they are basically clocks which shine light (ie radio signals) at observers from various places in the sky.
So please explain why you think you have insight into the universe when you have not actually examined the universe yourself, you're ignoring those people who have and you haven't got much, if any, understanding of the models of said people.
I'll suggest to you what I suggested to Jack (among other things), go learn how to construct and understand space-time diagrams in 1+1 dimensional special relativity. The choice in frames then can be expressed as different time slices of a worldline/light cone diagram. It'll make your mistake a bit clearer.
