“as far as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.”
Towards the end of the 19th century, mathematicians were in need of more abstract forms of mathematics, as the study of more complex and immeasurable objects like subatomic particles and computer science required it. Propositional logic was employed by philosophers to conceive of ways in which to conceptualize of infinitesimally small bits of matter and energy.
*Abstraction: the act of considering something as a general quality or characteristic, apart from concrete realities, specific objects, or actual instances.
Aristotle, the originator of empiricism and so many advances towards what would become known as the Scientific Method, defined Mathematics as “The Science of Quantity“. Today, there is no standard agreed upon definition of mathematics, as its level of abstraction has become so…well abstract. Three camps of mathematical philosophy currently predominate over the science today, The logicists, the intuitionists, and the formalists,
Mathematics is truly the continuing abstraction of objects (numbers, functions, sets..) from Natural whole numbers (1,2,3,4,5…) to parabolic geometry and beyond. The Formalists, like David Hilbert (Hilbert had a contestable argument for the development of General Relativity against Einstein) adopted Plato’s philosophy of forms in order to transform abstract concepts, into real abstract objects, making it possible to define subatomic particles as objects, though they cannot truly be observed or measured in the traditional Scientific sense.
An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing, an idea, or abstraction. (Abstracta: Lack Causation and Spatial Location.) This then would explain the need of Heisenberg’s Uncertainty Prinicple, which has been debated long before the advent of quantum mechanics and even before Socrates himself, in “Zeno’s Paradoxes”
Zenos Arrow Paradox:
In the arrow paradox,
Zeno states that for motion to occur, an object must change the position which it occupies.
He gives an example of an arrow in flight.
He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not.
*It cannot move to where it is not, because no time elapses for it to move there;
*it cannot move to where it is, because it is already there.
*In other words, at every instant of time there is no motion occurring.
*If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.
Many minds have tackled this and other of Zenos 9 remaining paradoxes, Aristotle explaining the paradox.
“If everything when it occupies an equal space is at rest, and
if that which is in locomotion is always occupying such a space at any moment,
Then: the flying arrow is therefore motionless.”
Aristotle, Physics VI:9, 239b5
My explanation for the paradox:
From General Relativity, we know that Time is Slowed by increasing Gravity, if we experienced Infinite Gravity then the arrow could actually become motionless, but in our (reality) environment we can only abstract time slowing, not actually slow it, to any appreciable extent.
Therefore, whenever we abstract a “what if” that cannot actually occur in “our real world”, we create paradoxes. Space-Time is fundamental to our experiencing of reality and required for our brain to process information linearly.
The First Problem
Suppose Homer wants to catch a stationary bus.
*Before he can get there, he must get halfway there.
*Before he can get halfway there, he must get a quarter of the way there.
*Before traveling a quarter, he must travel one-eighth; before an eighth,
*one-sixteenth; (1/64, 1/128, 1/256, 1/512, 1/1024, 1/2048, 1/4096, 1/8192, 1/16,384….)
This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.
The Second Problem
This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all.
Hence, the trip cannot even begin.
The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.
An alternative conclusion, proposed by Henri Bergson,
is that motion (time and distance) is not actually divisible.
This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts.
*Asymptotic: a line that approaches a given but does not meet it at any finite distance. When we halve the distance to infinity we never reach the goal. Natural Numbers (1,2,3,4,5..) are Integers (an Integer is a whole, and cannot be abstracted without losing its’ Essentiality) and are therefore indivisible.
*We cannot Abstract a Whole into its pieces and know anything about the whole. If one were to split a human, a car, an umbrella, an anything in half its essential character is gone and we have lost the meaning of our endeavor. Henry Bergson is correct.