The Greeks abhorred the infinite thinking that anything that was infinite was illogical. Zeno's famous paradox is a good example of how introducing infinity can cause logical trouble. Without allowing ourselves to use the infinite though we cannot arrive at calculus which uses the continuity of numbers as a necessary part of its operation. In some sense of course the Greeks allowed countable infinities grudgingly since they didn't think that there was a "last number" so to speak but allowing for unboundedness is a long way off from allowing the entity of infinity to exist as a philosophically analyzable entity. Allowing for unboundedness allows the Greeks at least partial access to the first order of infinity the countable infinities. For those of you who aren't terribly familiar with the concept of transfinite numbers there are different sizes of infinity the infinity that is the number of the counting numbers is called countable infinity because anything that is countably infinite can be put in a one to one correspondence with the counting numbers. That is countable infinities can be numbered while uncountable infinities are too large to be counted. If that doesn't make sense to you go find some material on cantor set theory.
The Greeks had a pretty good understanding of the rational numbers which are countably infinite. Saying that the Greeks were really only afraid of the real numbers (which are uncountable) is being too nice. They abhorred the existence of numbers like the square root of two since it could not be expressed as a rational number. The Greeks were quite happy to deal with the rational numbers as discrete objects but they would have frowned on talking about all of them as a whole. There is nothing disturbingly infinite about 2/3 but you cannot then haul off and start talking about all the numbers a/b that are between 0 and 1. The later the Greeks would have viewed as beyond the realm of logic because of its association with the infinite.
This inability to talk about questions which touched on the infinite or to use any sort of methods which invoked the infinite limited Greek mathematics. The reason of course for this abhorrence of the infinite goes deeper than the possibility of logical inconsistency, not everyone agreed with Zeno. The real reason is that in mathematics, logic, and geometry the Greeks did not just see disciplines of abstract thought. The profession of logician and geometer was to deal with the world as it really was. The Greeks took philosophy and mathematics as a way of uncovering deep truths about their own physical and metaphysical reality. The abhorrence of the infinite was rooted in the belief that the physical world cannot support the infinite and so any infinite argument or object does not have any reality and so does not deserve to be thought about.
I am afraid that in the intervening time physicists may have become afraid not of the infinite but of the finite. Mathematicians also though to a lesser extent tend to distance themselves from problems that are inherently discrete. We have learned how to deal with the infinite in a logical and rigorous way and we have found the tools that this dealing with the infinite has given us so useful that we have become at least partially unable to work without them. The assumption of the continuity of space is so basic to the world of physics that is essentially completely unchallenged in all of physics. Even in quantum mechanics where things become discrete instead of continuous the underlying space is still thought to be continuous. String theory is even worse since it assumes a completely flat subspace in which the strings are to interact. If you don't see the connection between continuity and the finite then I will just take a moment to point out that continuity is a concept that cannot be achieved without the use of something similar to the real numbers and that means the introduction of uncoutable infinities. Countable infinities are what you get when you allow discrete systems to be unbounded. So intrinsically countable systems are discrete. Since countable systems are necessarily discrete (even though any two rational numbers can approach each other as close as you like every set of rational numbers is disconnected)then that implies that countable systems are necessarily made up from finite objects.
What I really am talking about when I say that physicists have a fear of the finite is that they have a fear of methods that are based in discrete objects. Calculus with its dependence on uncountable infinity and the continuity of things that are being examined is totally indispensable to modern physics. I don't necessarily think this is a bad thing in the sense that calculus is a wonderfully robust and interesting tool and we should be not afraid to use it. However physicists should have a way around calculus no matter how painful and difficult there ought to be a way to do physics that is fundamentally discrete but there is no such thing. Mathematicians and computer scientists can play with the discrete but physics remains fearful of it.