It is possible that I am simply going to repeat myself, or at least state things which most people would say ‘Of course’ to. On the other hand, sometimes stating the blinking obvious is a good thing to do.
I suppose the first thing I want to suggest is that things, generally, go in fashions. There might be, for example, a fashion dressing your soldiers up in fancy red uniforms to march into battle. There might not, in fact, be any particularly good or bad reason for doing this. It might be that red is a martial colour in your culture, and thus seems to be a good option for trying to instil some valour, ardour or some such into the troops. Your enemy might have a different culture, wherein, say, red is the colour of cowardice and blue is that of valour. They might, thus, be very willing to attack your red clad troops, but flee from the odd blue bedecked regiment of cavalry that you have permitted just for a spot of variation.
Fashions, however, change. These changes can be brought about by a variety of causes. For example, a fair bit of European army kit changed when those armies were fighting in places that were not Europe. Light infantry, irregulars, and different sorts of swords and so on were all incorporated into military equipment and tactics as a result of experience beyond the home shores of the culture. This is, of course, not unique. The Romans pinched their sword from the Spanish, after all.
Fashions are more than simply in dress, equipment or tactics. I have very briefly tried to suggest that ideas have an impact on how warfare is conducted. The particular example I had in mind was the ‘star’ fortress style, particularly associated with Vauban but used extensively before then. The idea is of course usually explained fairly simply as a response to the increased firepower of cannon in besieging armies. The instrumental requirements of a form of fortification which could resist cannon fire more effectively led designers to build lower walls, packed with earth, and with open fields of fire. A round tower simply could not match the modern fortress in ability to block the attackers and blunt their force.
Except, of course, that often old style castles could, and did, resist modern siege weapons. One problem, at least, facing would be besiegers, was that of actually getting the siege weapons into place. This was a huge logistical problem. Once arrived, there was the problem of siting the weapons so they could be effective and, of course, of ensuring the supply of the besiegers, in terms of food, water, weapons and men. A siege of even a simply medieval style castle was no joke, as some of the problems encountered in the English Civil war demonstrate.
But another issue in the design of castles must have been that of the intellectual changes brought about by the renaissance. While medieval castle builders did, indeed, use mathematics to calculate, for example, how many steps were needed in a spiral staircase, star style fortresses needed some more advanced ideas about geometry. Indeed, if these were not forthcoming, there arose some slightly odd shapes. For example, the defences of Berwick on Tweed are irregular. This might be due to the lie of the land, to the inadequacy of surveying or to someone who was a bit dangerous to let loose with a compass and set square. Of course, one might suggest to Her Majesty that that was how it was meant to be all along, and it is perhaps fortunate that the defences were never seriously tested, but the idea was there of something pleasingly geometric in shape, and that comes from the interest in mathematics of the renaissance.
Of course, warfare is not the only place where one sees this. Galileo was interested in geometry, and there is evidence that his concepts were based on the idea of geometry. Spinoza, too, thought geometrically, as least, insofar as his Ethics is written like a geometry text book. Euclid was viewed as the ultimate in comprehensive proof, and people attempted to emulate him. The idea of geometry, thus, permeates a fair bit of intellectual culture in the sixteenth and seventeenth centuries, and it would be a rash historian indeed who suggested that it did not affect military theory and practice.
We can possibly push on a little further with this notion. The military text books of the day were rather obsessed with order and geometry, with nice diagrams explaining how to array a certain number of men in blocks so many deep and thus how wide they should be. Montecuccoli, for example, suggests an octagonal infantry battalion at one point, for battalions at the ends of the battle line. This is a sophisticated sort of formation, and it indicates, from the text, that he is interested in geometry, and would seem to me to need seriously trained soldiers to carry it out without getting into severe disarray.
The point is that Montecuccoli, as a theoretician as well as a general, we thinking geometrically, as part of the culture in which he was educated. To think geometrically is simply to be part of the intellectual culture of the day. That is not to say it was used uncritically, although that accusation could be aimed at some of the text books. It was simply part of how things were thought about.
There were reverse influences, as well. Asking a theoretician to design a fortress could have impact on the development of mathematics, geometry and architecture. I seem to recall that Durer was one such, although I think he pre-dated the star style fortress. Nevertheless, he can be seen as a transitional figure, designing shorter, stubby towers packed with earth to resist cannon fire, an idea taken up by Henry VIII in the defence of English shores.
I was going to start talking about the influence of ideas on tactics, such as the caracole, but I am starting to think that this piece is both long enough and both arguable enough and not radical enough, so I shall have to leave it there and return to the Curious Case of the Caracole another time.