Huw Price and Ken Wharton in Aeon:
[T]hat one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to the other, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it.
On the other hand, there’s his own theory of gravity, published in hisPrincipia several years earlier. It says that one body can exert a force on another, at arbitrary distance, without the need for any intermediary. What was a poor genius to do?
How Newton dealt with this dilemma in his own mind is still a matter for debate. Privately, his letter to Bentley continues: ‘Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left open to the consideration of my readers.’ In public, he seems to express disdain for the question: ‘I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses.’
Two centuries later, Albert Einstein got Newton off the hook – though not before he’d made the problem even worse. Einstein’s 1905 theory of special relativity raised a new difficulty for Newton’s theory of gravity. Instantaneous action-at-a-distance requires that the distant effect is simultaneous with the local cause. According to special relativity, however, simultaneity is relative to the observer. Different observers disagree about which pairs of events are simultaneous, and there’s simply no fact of the matter about who is right.
Without simultaneity at a distance, the notion of instantaneous action-at-a-distance doesn’t make sense.
More here.