[...] For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3… On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever.
This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not.
At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants. Read more
And Paul Blackham in recent comments, speaks of the mode of enquiry that drove Galileo and Francis Bacon:
Galileo’s notebooks... are not full of the rigorous, hard-nosed observational data that the mythology depicts. In fact, he can’t see the things that he is convinced he could see if he had better telescopes. Some of the drawings of what he sees through his telescope do not support his arguments. He marvels that Copernicus persisted with his argument even when his observations were so inaccurate. When we compare Galileo’s drawings of the moon with photographs of the moon, it is hard to find similar features. The point is that Galileo was FIRST convinced of the heliocentric view and then began to develop telescopes that would enable him to observe what he was convinced was there. Kepler who wanted one of these new telescopes was disappointed by the results. He found them to be accurate for earthly observations but misleading for heavenly. Yet, the quality of the observations was not the critical factor here. It was the development of a new paradigm for viewing the cosmos, one whose benefits were only unfolded as time went on.
In Galileo’s letter to Leopold of Toscana of 1640, he specifically says “I am unwilling to compress philosophical doctrines into the most narrow kind of space and to adopt that stiff, concise and graceless manner, that manner bare of any adornment which pure geometricians call their own, not uttering a single word that has not been given to them by strict necessity…”.
In other words, Galileo knows full well that his argument is not a matter of pure observation [whatever that may mean] but a philosophical perspective first. Francis Bacon, whose scientific arguments were so vital to the foundation of the entire tradition, argues that we need to view the world with “unbiased senses” – by which he means that our senses need to be rebuilt with a new way of perceiving that mirrors the world rather than ourselves – “For man’s sense is falsely asserted to be the standard of things; on the contrary, all the perceptions, both of the senses and of the mind bear reference to man and not to the universe, and the human mind resembles those uneven mirrors which impart their own properties to different objects from which rays are emitted and distort and disfigure them.” [Novum Organum, Aphorism 41]. He speaks of a need to demolish the way we think and perceive so that a new way of seeing/thinking can be built. In the preface to the Novum he says “Our only hope of salvation is to begin the whole labour of the mind again… after having cleansed, polished and levelled its surface.” Preconceived notions, opinions and even common words all need to be “renounced with firm resolution… so that access to the kingdom of man, which is founded on the sciences, may resemble that to the kingdom of heaven, where no admission is conceded except to children.”
Copernicus wrote in the preface to ‘De Revolutionibus’ that the astronomical tradition of Aristotle could only solve the classic problems with great complexity and that a new paradigm was needed.
The point of all this is to simply note that our philosophical/theological convictions do not only shape and colour our observations, but they also determine what and how we observe. There is no escape into geometry or any other simple observation/calculation that is free from the theological and philosophical arguments.