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Are Infinity Machines conceptually possible?

While infinity is commonly defined as ‘the unlimited, that which goes beyond any

fixed bound’1, Max Black uses two different working definitions of infinity in order to

suggest the antinomies of Zeno’s Paradoxes. For Black’s argument, infinity is both a

physical extremity (that which is infinite in extent) and divisibility (that which is

infinitely divisible). Black describes a series of machines which by their very nature are

contradictory in order to explain the common misconceptions of philosophers when

attempting to solve Zeno’s paradox of Achilles and the Tortoise.

Firstly, Black conjures up a machine called Alpha, whose task it is to move an

infinite number of marbles from a tray on its left to a tray on its right. He states that as

Alpha continues with its task, the time it takes for the action to be completed continues to

reduce itself by halves (that is, it takes Alpha at first one minute to move the first marble,

and then half a minute for the second one, and so on infinitely). Because Alpha’s speed

increases as its task continues, it should, after four minutes, have moved all the marbles

from the left to the right. But how can we be sure that Alpha counted an infinite number

of marbles if the task is completed after only four minutes? How can there be a finite

time for counting an infinite sum?

Black’s next machine, Beta, functions on exactly the same principle, but in order

to prove that the marbles counted are infinite, this machine only counts one marble over

and over again. This in itself is contradictory, Black states, because the marble is always

returned to its starting point, and therefore no progress is made. He asserts that Beta is

therefore performing a logically impossible and self defeating operation in moving the

marble from left to right, only to find it on the left tray again. That is, for every time the

marble is moved to the right tray, it must and will be returned to the left; but for every

time the marble is returned to the left, it must and will be moved to the right. The

problem with Black’s assertion of this machine being logically impossible lies in his

definition of infinity. Black is, in this case, stating that it is logically impossible because

1 Oxford Dictionary of Philosophy

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the machine is constantly taking one step forward and one step back, thus remaining

stationary. This is only because Black is assigning this task the definition of infinity

which suits his argument. As mentioned earlier, infinity is both a physical extremity, and

divisibility. Therefore, it does not matter if the machine is making any directional

progress in counting the same marble, it only matters that the machine is counting the

marble infinitely.

Black’s third machine, Gamma, is said to function in the same way as Beta, but

while Beta moves the single marble from left to right, Gamma moves it from right to left,

constantly replenishing Beta’s store, and vice versa. Herein lays the problem: for Gamma

to succeed, Beta must fail, and for Beta to succeed, Gamma must fail. The marble cannot

be both on the left tray and on the right tray at the end of the four minutes. If the marble

is on the right tray, then Beta has succeeded in counting an infinite number of marbles,

but Gamma obviously hasn’t, as it still has a marble to count and move. The same works

if the marble is on the left tray, and so we see that Gamma and Beta being assigned the

same tasks but in opposite directions poses a contradiction.

Black’s next two machines, Delta and Epsilon, function exactly in the same way

as Beta and Gamma do, and are used by Black to illustrate that the time intervals are

irrelevant for the success or failure of the machines. Delta and Epsilon find themselves at

the same directional contradiction as Beta and Gamma; for one to succeed the other must

fail, and therefore, according to Black, neither can succeed. But why can neither succeed?

If one machine succeeds and the other fails, there is still a 50% success rate between the

two machines, and at the end of the exercise, an infinite number of marbles have been

counted, regardless of one machine’s failure. Perhaps it is in the nature of infinity

machines to have one fail in order for the other to succeed, thus making them a logical

partnering of devices that still serve their function accurately.

Black’s final machine, Phi, functions in the same manner as Alpha and Beta, but

with the marbles it transfers becoming geometrically smaller as they are counted, so that

eventually at the end of the four minutes, the marble will be shrunk to nothing, and

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therefore eliminating the problem with its final location. Black’s only way of proving that

this machine is conceptually impossible is by asserting that if all other infinity machines

are impossible, then Phi must be as well, as there would be a logical inconsistency to

have one machine logically possible and the others not. Since Black believes he has

proven all the other machines to be logically impossible as they cannot succeed and fail

at the same time, Phi must also be impossible.

It can be argued, however, that Black’s infinity machines are not logically or

conceptually impossible, but rather, that they are not considered in the right parameters in

order to promote their possibility. With Alpha, the impossibility lies in counting an

infinite sum in a finite time. But that is simply a matter of poorly defining ‘infinity’.

Alpha functions using the two working definitions of infinity as interchangeable: the

infinite amount of marbles Alpha is counting refers to infinity as a physical extremity: the

marbles are infinite in extent. The time constraint refers to infinite in terms of divisibility,

as the time is constantly divided smaller and smaller. It could be argued that the machine

will not be finished after four minutes; but rather, that like in Zeno’s paradox, it will

never be finished, as the time continues to reduce its self into smaller and smaller

denominations, but never reaches zero. This proves Alpha is not logically inconsistent,

but that Black has not considered the infinite divisibility of time in proportion to the

infinite amount of marbles being counted.

Beta and Gamma, as well as Delta and Epsilon are also not inconsistent. As single

entities they function to count the marble over and over again, an infinite amount of time.

When placed against each other so that one must fail in order for the other to succeed,

still a success is being achieved. It could be argued that once the machines are placed so

that they function together (one replenishing the other’s marble and vice versa) they

cease to be separate entities and become connected and functioning as one machine (or a

system). If this is true, then there is no failure as they are functioning together as a means

to an end. As long as the marble is found on either the left or the right, the system, or

larger machine, has been successful in counting the marble an infinite number of times.

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Phi, Black’s final machine has no flaw in it other than Black’s assertion that if

none of the other machines are possible, then no infinity machine can be possible. But

since I have just proven that all of Black’s infinity machines are indeed conceptually

possible in their own right, Phi as well is possible.

To conclude, while infinity machines are conceptually possible as machines

which count an infinite spread of marbles over a finite period of time, it is conceptually

impossible to count an infinite sum in a finite period of time. So, regardless of the logical

consistency of the infinity machine, there still remains a logical inconsistency in the

fundamental principle. There simply isn’t enough time to count an infinite number of

anything, therefore all the machines would continue counting well past the four minute

mark, as they would continue to divide the time into smaller denominations infinitely. If

not, they would be successful until just past four minutes, when it becomes clear that

there has not been an infinite amount of marbles counted. But that does entertain the

notion that up until just past four minutes, the machines have been successful in counting

an infinite sum and only in hindsight is it apparent that the machines failed. It could also

be argued that if such a machine can be conceived, then it must be conceptually possible,

else we would have not been able to conceive it (similar concept to the Theist’s

Ontological Argument). But the catch is simply this: the outcome of the machine’s is

completely independent of what the machine’s task was prior to that outcome; that is, the

end and the means are independent of each other, so the machines’ existences should not

be determined by their success or failure in counting the marbles, only in whether or not

they are capable of counting an infinite sum.