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Mathematics/BigelowVsField: can be understood realistically when viewed as a study of universals, properties and relations, of patterns and structures of things that can be in different places at the same time.
Mathematics/Realism/Bigelow/Pargetter: Pro Realism of Mathematics.
((s) The thesis that numbers exist as objects. And thus also sets, and all possible mathematical objects or entities. (FieldVs.)
We agree with the antirealists that there are human creations:
For example, words, ideas, diagrams, images, terms, theories, texts, academic departments, etc.
Realism/Bigelow/Pargetter: of mathematics: is well compatible with modal realism.
Science/Bigelow/Pargetter: no one believes that everything in science is real. There must be (useful) fictions. Therefore, one can in principle be a realist in relation to everyday things and at the same time a mathematical antirealist. For example, Field:
Field/Bigelow/Pargetter: is at the same time a realist regarding space-time, particles and fields.
Realism/Antirealism/Mathematics/Bigelow/Pargetter: nevertheless, there is something wrong with this marriage: mathematics is not a small element of science but a very large one. It is also not easy to isolate. Example:
Galileo/Bigelow/Pargetter: did not know about instantaneous speed yet. For him, speed was simply a course divided by time. A falling object then had an average speed, although Galileo was not aware of this either.
Therefore, he made the following mistake: if two bodies are dropped together and one of them continues to fly, they both have exactly the same speed until the first one stops.
Galileo: but had to assume that this body was slower, because the other body needed less than twice as much for the eventual double distance.
Rate of fall/Bigelow/Pargetter: therefore the average velocity cannot be proportional to the distance.
Realism/Bigelow/Pargetter: if anything is evidence for realism, it is this: an object that falls twice as far does not have twice the average velocity. If you find out, you are a realist in terms of how long it takes for an object to reach a given distance. This makes us realists in terms of velocity, time and distance.
((s) The problem arose from the fact that Galileo was forced to adhere to the definitions he had set up himself, otherwise he would have had to change his theory.).
Average/VsRealism/Bigelow/Pargetter: one could argue that average is only an abstraction.
VsVs: we do not need the average here at all: it is simply true that the object falls faster in the second section, and that simply means that the average velocity cannot be the same.
Velocity/Galileo/Bigelow/Pargetter: he respects that it is physically real. And caused by forces and proportional to these forces, so velocity was causally effective for him.
Velocity/today/Bigelow/Pargetter: we think today that it is the instantaneous speed which is causally effective, never the average velocity.
Realism/Mathematics/Bigelow/Pargetter: the equations we use to describe the relations between different falling objects are human inventions, but not the relations themselves.
Rate of fall/fall law/Galileo/Bigelow/Pargetter: the distance is proportional to the square of the time traveled. How is this abstract law based on concrete physical facts?
Galileo: in the first unit of time the body falls a certain distance, in the second unit not double, but triple of this distance, in the third five units, and so on.
Predecessor/Bigelow/Pargetter: this had already been anticipated in the Middle Ages.
Middle Ages/Thesis: an increment has been added to each section. One, three, five, seven...
Now the sum of the first n odd numbers is n².
Then it seems to be based on nothing but rules for the use of symbols that
(1 + 3 +... + (2n - 1) = n².
But this is a mistake:
Numbers/Number/Bigelow/Pargetter: may be abstract, but they are present in an important sense in the physical objects: in a collection of objects that have this number, they are the common thing. For example, a collection of objects which has the number n².
You can just see that the pattern has to go on like this.
And so it is in Galileo's case.
Realism/Mathematics/Bigelow/Pargetter: the differences to physical bodies should not blind us for the similarities. If objects instantiate the same numbers, the same proportions will exist between them. (>Instantiation).
Instantiation/Bigelow/Pargetter/(s): For example, a collection of 3 objects instantiates the number three.
Equation/Bigelow/Pargetter: (e.g. Galileo's fall rate, which was wrong) is a description of real relations between real objects.
Platonism/Bigelow/Pargetter: this view can roughly be called Platonist.
Bigelow/Pargetter: pro Platonism, but without the usual Platonic doctrines: we do not assume forms or ideals taken from an earlier existence that we cannot see in our world, and so on.
Realism/Universal Realism/Universals/Bigelow/Pargetter: our realism is closer to Aristotle: the universals are here in our world, not in an otherworldly.
BigelowVsAristotle: we disapprove of his preference for quantitative versus quantitative characteristics of objects.
Patterns unfold patterns. The structures of mathematics show up not only in the hardware of physics, but also in the "mathware", through properties and relations in different areas of mathematics. For example, not only objects, but also numbers can be counted. Proportions, for example, stand in proportions to each other. This is the reflexivity within mathematics._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990