ediates,-a substance which is neither number nor points nor spatial magnitude nor time. And if this is impossible, plainly it is also impossible that the former entities should exist separate from sensible things.
And, in general, conclusion contrary alike to the truth and to the usual views follow,Belstaff XL500 Replica Jackor, if one is to suppose the objects of mathematics to exist thus as separate entities. For because they exist thus they must be prior to sensible spatial magnitudes, but in truth they must be posterior; for the incomplete spatial magnitude is in the order of generation prior, but in the order of substance posterior, as the lifeless is to the living.
Again,Alex Ovechkin Tröjor, by virtue of what, and when,Luca Sbisa Tröjor, will mathematical magnitudes be one? For things in our perceptible world are one in virtue of soul,Thomas Greiss Tröjor, or of a part of soul, or of something else that is reasonable enough; when these are not present,Matt Cullen Tröjor, the thing is a plurality,Parajumpers Dameklær Jakke, and splits up into parts. But in the case of the subjects of mathematics, which are divisible and are quantities,Curtis Joseph Tröjor, what is the cause of their being one and holding together?
Again, the modes of generation of the objects of mathematics show that we are right. For the dimension first generated is length, then comes breadth, lastly depth, and the process is complete. If, then, that which is posterior in the order of generation is prior in the order of substantiality,NHL Mens Buffalo Sabres Black Camo Stack Pullover Hoodie, the solid will be prior to the plane and the line. And in this way also it is both more complete and more whole, because it can become animate. How,Belstaff Coral Vent Jackor, on the other hand, could a line or a plane be animate? The supposition passes the power of our senses.
Again, the solid is a sort of substance; for it already has in a sense completeness. But how can lines be substances? Neither as a form or shape,NHL Mens Detroit Red Wings Black Rink Warrior Pullover Hoodie, as the soul perhaps is,NHL Mens New York Islanders Black Rink Warrior Pullover Hoodie, nor as matter,Canada Goose Lodge Down Hoody Jackor, like the solid; for we have no experience of anything that can be put together out of lines or planes or points, while if these had been a sort of material substance, we should have observed things which could be put together out of them.
Grant, then, that they are prior in definition. Still not all things that are prior in definition are also prior in substantiality. For those things are prior in substantiality which when separated from other things surpass them in the power of independent existence, but things are prior in definition to those whose definitions are compounded out of their definitions; and these two properties are not coextensive. For if attributes do not exist apart from the substances (e.g. a ‘mobile’ or a pale’), pale is prior to the pale man in definition, but not in substantiality. For it cannot exist separately, but is always along with the concrete thing; and by the concrete thing I mean the pale man. Therefore it is plain that neither is the result of abstraction prior nor that which is produced by adding determinants posterior; for it is by adding a determinant to pale that we speak of the pale man.
It has, then, been sufficiently pointed out that the objects of mathematics are not substa
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