1.) A hypothesis about mathematical abstraction: idealized objectivity of what is sensed.
2.) My standard for objectivity of the sensed is what all possible sentient beings must sense when sensing the same object. We recognize that different animals have the same organs and/or sense the same things but have different experiences. We see a leaf as a single color while a bee sees it as two; sounds that are inaudible to us are deafening to dogs; temperatures that are freezing to us are oppressively hot for some deep-sea fish, etc. So it seems like one good account of mathematical abstraction is the quantitative reality that must be the same in all these possible experiences, whether we’re talking about the objectively present shape (geometry), or the relation between parts and their common measure (numbers).
3.) This quantity we come to is neither is large or small, nor set off from its background by some contrast of color (like black lines on white paper). It is a quantity of almost pure touch – a triangle drawn on one’s back in a dark room, and considered as neither large nor small, or three taps on the back, which need not be hard or soft, hot or cold, or of any determinate size or location.
4.) Our standard for the objective includes motion too, but this motion can’t arise out of the quantity. Motions need to be continuous but changes in idealized quantity don’t. Doubling a line does not require us to imagine it 50% longer first, and adding sides to regular polygons so as to approach a circle can happen by jumping from equilateral triangle to square to pentagon to hexagon, etc.
5.) This “jumping” requires two things (1) space, i.e the background of the shapes, or undifferentiated extension with no definite form, but also (2) the positing of something that stays numerically one while jumping from triangle to square to pentagon, etc. These two are completely different: it’s not as if undifferentiated extension jumps from one shape to another – this would be to say that what stays the same between the shape and its background does not stay the same.
6.) But it seems that all there is to a shape is its fixed border and the definite area of space that this gives rise to; or a definite amount of space that might allow for different fixed borders. So there is simply no such thing that remains numerically one while the line is extended or “the figure” jumps to “its” limit. Change in quantity is only intelligible for one who assumes – whether knowingly or not – some source of unity that quantity does not have. It is ironic that we call this numerical unity since quantity cannot possess it of itself.
7.) But even after we assume some source of numerical quantity allowing for change, there is still no necessity for the change to pass through intermediate stages. If it did, taking a figure to a limit or superimposing one figure on another would take time for the same reason that growing a plant or crossing the room takes time. And so when we idealize the objectivity of the sensed in mathematical quantity, we lose the necessity that motion be continuous, both over its parts and over time. Thus the passage or flow of time will be lost, and we will be left with only its difference, i.e. this point of time is not that one. But nothing whatsoever will connect these two points, for to do so it would have to be continuous and this is precisely what is denied. It follows that there cannot even be two points at different times. And so just as we had to assume a source of numerical unity which cannot belong to the quantity as such, we have to assume a continuity of time that cannot belong to quantity. It makes no difference whether this continuity is seen as belonging to time itself, or to motion first and then to time.
8.) Both change and quantity must be common to all animal experience, and so both meet the criteria for being objective. But we can’t preserve the reality of change under the assumption that this idealized objectivity gives us nothing but quantity.
9.) And so an idealized, objective world can’t be achieved in a single sort of abstraction but in the abstractions giving us the mathematical, motion and time, and the source of numerical unity. The objective account of the sensible world will divide into the mathematical, physical, and metaphysical sciences. That said, mathematics takes pride of place for the exactitude and clarity of the knowledge we get from it.
10.) Mathematics thus enters physics to the extent that the physicist wants to understand the sensible world in a mode excelling all others in exactitude and clarity, and is willing to accept the trade off of losing sight of the passage of time and the reality of change. There is no miracle of math working – it is simply an idealized sense world, and even then one that comes with trade-offs in our understanding of nature.
10b.) Calling math an idealization explains why it can both explain the world while frequently looking nothing like the world we sense. Like any idealization of X, it will have properties that are illustrative of X even when they are impossible for X. The same property can harmonize why Plato thought mathematics formed an ideal world subtending all objectivity while St. Thomas thought it was an abstraction from that world.
11.) We’ve had a good deal of success with the dialectical unification of math and physics. Even here, however, we inevitably lose the reality of time and change, and, a fortiori, we become indifferent to the source of numerical unity. If nothing stays numerically one throughout change, then there can be no causality of any kind. There is clearly no material or agent causality, and so neither are there any causes existing in relation to them. The attempt to limit our account of nature to the perfectly objective not only overlooks something perfectly objective, but end up making it impossible.