Think of how many generations of humanity would have benefited if certain ideas had been invented sooner, rather than later - if the Greeks had invented science - if the Romans had possessed printing presses - if Western civilization had turned against slavery in the thirteenth century.
Archimedes of Syracuse was the greatest mathematician and engineer of the ancient world. Imagine that Archimedes invented a temporal telephone ("chronophone" for short) which lets him talk to you, here in the 21st century. You can make suggestions! For purposes of the thought experiment, ignore the morality of altering history - just assume that it is proper to optimize post-Archimedean history as though it were simply the ordinary future. If so, it would seem that you are in a position to accomplish a great deal of good.
Unfortunately, Archimedes's chronophone comes with certain restrictions upon its use: It cannot transmit information that is, in a certain sense, "too anachronistic".
You cannot suggest, for example, that women should have the vote. Maybe you could persuade Archimedes of Syracuse of the issue, and maybe not; but it is a moot point, the chronophone will not transmit the advice. Or rather, it will transmit the advice, but it will come out as: "Install a tyrant of great personal virtue, such as Hiero II, under whose rule Syracuse experienced fifty years of peace and prosperity." That's how the chronophone avoids transmitting overly anachronistic information - it transmits cognitive strategies rather than words. If you follow the policy of "Check my brain's memory to see what my contemporary culture recommends as a wise form of political organization", what comes out of the chronophone is the result of Archimedes following the same policy of looking up in his brain what his era lauds as a wise form of political organization.
You might think the next step would be to prepare a careful series of Plato-style philosophical arguments, starting from known territory, and intended to convince an impartial audience, with which to persuade Archimedes that all sentient beings should be equal before the law. Unfortunately, if you try this, what comes out on Archimedes's end is a careful series of Plato-style philosophical analogies which argue that wealthy male landowners should have special privileges. You followed the policy of "Come up with a line of philosophical argument intended to persuade a neutral observer to my own era's point of view on political privilege," so what comes out of the chronophone is what Archimedes would think up if he followed the same cognitive strategy.
In Archimedes's time, slavery was thought right and proper; in our time, it is held an abomination. If, today, you need to argue that slavery is bad, you can invent all sorts of moral arguments which lead to that conclusion - all sorts of justifications leap readily to mind. If you could talk to Archimedes of Syracuse directly, you might even be able to persuade him to your viewpoint (or not). But the really odd thing is that, at some point in time, someone must have turned against slavery - gone from pro-slavery to anti-slavery - even though they didn't start out wanting to persuade themselves against slavery. By the time someone gets to the point of wanting to construct persuasive anti-slavery arguments, they must have already turned against slavery. If you know your desired moral destination, you are already there. Thus, that particular cognitive strategy - searching for ways to persuade people against slavery - can't explain how we got here from there, how Western culture went from pro-slavery to anti-slavery.
The chronophone, to prevent paradox, will not transmit arguments that you constructed already knowing the desired destination. And because this is a law of physics governing time travel, the chronophone cannot be fooled. No matter how cleverly you construct your neutral-sounding philosophical argument, the chronophone "knows" you started with the desired conclusion already in mind.
The same dilemma applies to scientific issues. if you say "The Earth circles the Sun" it comes out of the chronophone as "The Sun circles the Earth". It doesn't matter that our civilization is right and their civilization is wrong - the chronophone takes no notice of facts, only beliefs and cognitive strategies. You tried to transmit your own belief about heavenly mechanics, so it comes out as Archimedes's belief about heavenly mechanics.
Obviously, what you need to transmit is the scientific method - that's how our own civilization went from geocentrism to heliocentrism without having the destination already in mind. Unfortunately, you also can't say to Archimedes, "Use mathematical laws instead of heroic mythology to explain empirical phenomena." It will come out as "If anyone should throw back his head and learn something by staring at the varied patterns on a ceiling, apparently you would think that he was contemplating with his reason, when he was only staring with his eyes... I cannot but believe that no study makes the soul look on high except that which is concerned with real being and the unseen." (Plato, The Republic, Book VII.) That is Archimedes's culture's stance on epistemology, just as science is your own culture's stance.
Can you suggest that Archimedes pay attention to facts, and authorities, and think about which one should ought to take precedence - by way of leading him down a garden path to the scientific method? But humanity did not invent the scientific method by setting out to invent the scientific method - by looking for a garden path that would lead to the scientific method. If you know your desired destination, you are already there. And no matter how you try to prevent your garden path from looking like a garden path, the laws of time travel know the difference.
So what can you say into the chronophone?
Suppose that, at some point in your life, you've genuinely thought that the scientific method might not be correct - that our culture's preferred method of factual investigation might be flawed. Then, perhaps, you could talk into the chronophone about how you've doubted that the scientific method as commonly practiced is correct, and it would come out of the chronophone as doubts about whether deference to authority is correct. After all, something like that must be how humanity got to science from nonscience - individuals who genuinely questioned whether their own culture's preferred method of epistemological investigation was correct.
If you try to follow this strategy, your own doubts had better be genuine. Otherwise what will come out of the chronophone is a line of Socratic questioning that argues for deference to authority. If your doubts are genuine, surface doubts will come out as surface doubts, deep doubts as deep doubts. The chronophone always knows how much you really doubted, and how much you merely tried to convince yourself you doubted so that you could say it into the chronophone. Such is the unavoidable physics of time travel.
Now... what advice do you give to Archimedes, and how do you say it into the chronophone?
Addendum: A basic principle of the chronophone is that to get nonobvious output, you need nonobvious input. If you say something that is considered obvious in your home culture, it comes out of the chronophone as something that is considered obvious in Archimedes's culture.
Rereading this article from Emile's link:
I think my major problem with this article is that the perfectly reasonable conclusion - that you can't do better in the future today by thinking today's cached thoughts about how people in the past could have done better in the future of the past - is obscured by the utterly ridiculous device of the eponymous chronophone.
To elaborate: If you analyze the question, "how did people in the past figure out that ideas are tested by experiment?", then you can immediately rule out "by asking how people at their own time evaluated ideas". And indeed, generalizing to "how did people in the past become smarter than their contemporaries - i.e. better at solving their problems?", you see that it is trivially true that you can't count on the standardized thinking of the present to take you beyond the standardized thinking of the present.
But if you analyze the question, "what do you say into the chronophone to convince Archimedes?", you come up with "this thing couldn't possibly work - there is no way to draw a unique relation between rationalizations of current and past ideas, so it fails sci-fi". Which has nothing to do with anything.