Mainly, I presume, on the part of myself. — Wayfarer
Certainly, congratulations, but I think it is, also, more perfectly general, almost a universal disease.
I am prolix, due to some clarity, ergo, I try to see this all more clearly through the survey which does not yet acquire the full masterly pinnacle of vantage over the material in question.
“that we concur on is a number of objects, but the number cannot be said to exist independent of its apprehension, at least, not in the same way objects apparently do.”
This is not the same issue. That is the current way of using numbers. What the Greeks say is: there is no thing that corresponds to the number. Counting numbers are not units. Units are only in the mind. There is not one apple, since it is a different object in the region of change, than another thing that is the same. Same doesn’t mean the same thing as equal. Even if the atoms of the whole apple were of the same counting number. Same can mean, belongs together, and it can mean somehow an “identity”, though, strictly speaking, the only identity is with an apple and itself. Another shares in the same form or idea, or genus.
The unclarity here lies in the radical difference the ancients gave to heavenly motion, and earthly. And its transitional phase in Galileo, with half a foot in the medieval, "astronomy is math in time and place". And another moving towards the modern view.
“So numbers are not 'objective' in the same way that 'things' are.”
The Greeks don’t have a notion of objective as what is not dependant on the human being. What they mean is that the intellect is literally a manner of perception. It is ontological. Speaks of the substance. Modern physicists, apart from a few like Penrose, don’t regard maths that way.
“I can't understand how this could be true, as so much of Galileo's fame rests on precise measurement. How can measurement of celestial objects or mass not depend on applied mathematics?”
Because they regard it as a claim about what the appearances are. The truth about the phenomena. Hypothesis used to mean throwing something under, into the substance. Notice the prefix, hypo. A putting under in the supposatium or foundation. The word hypothesis has changed meaning, now it means a working methodological consideration that wants to bring forward a result. It works! Success! that’s why I used the Rutherford example above, to point to this change. There was no, from a doctrinal point of view, I don’t speak de facto, applied mathematics in former times.
Measurements are tools in our (modern) usage, which is why we have maths of approximation. It’s simply a gauge of the rigour of the science so far as it can bring out precice results. It’s linked to practice, not to ontological knowledge. Remember the word of Feynman, rigour in the sciences is the number of decimal points. He says, also, of concepts in physics, it must do something, only then do we know it exists.