Irresistible force paradox as an inconsistent formal theory

Question

The irresistible force paradox asks the following question:

"What happens when an unstoppable force meets an immovable object?"

I ask myself the following question: we could construct a system of axioms asserting the existence of forces, motions and objects, in which it would be possible to interpret the proposition that there exists an irresistible force (which always produces motion) and that an immovable object (which is never moved independently of the force). And show that this system of axioms is in fact inconsistent because it is impossible to find a model that satisfies the axioms? Would that be a way to show that this ancient paradox is in fact self-contradictory?

Answer 1

This dilemma doesn't really makes sense in modern physics because any macroscopic force accelerates any mass no matter how large, but if you were to axiomatize it in pre-modern physics then yes, it would be inconsistent. You would end up with a force F such that for all objects x, F moves x, and an object X such that for all forces f, f does not move X. You could then derive the inconsistent proposition: F moves M and F does not move M.

However, this isn't really a paradox; it's a dilemma to be used against the concept of omnipotence. For example, if God can do anything, can he make a rock so big that he can't lift it? This sort of dilemma forces the theist (if he wishes to be logically consistent) to limit what is meant by "omnipotent".

It is easy to come up with such dilemmas: Can God make a blade so sharp that it can cut anything and a string so tough that it can't be cut? Can God make an acid that can dissolve anything and a flask to hold it that can't be dissolved? If you don't have a concept such as omnipotence to justify postulating the two opposing things, then it's just a word game: "What if something can be done and also can't be done?" In this case the dilemma is not philosophically interesting.

Answer 2

The concepts involved have infinities hiding in them. "Irresistable" means the thing has an arbitrarily large ability to overcome resistance. "Immovable" means it has an arbitrarily large ability to provide resistance.

Mathematically, our primary method of dealing with infinity is as a limit. Zeno's paradox, for example. The famous runner has to cross half way, then half of the remaining distance, then half of the new remnant, and so on. We deal with this by constructing a sequence.

N=1 : 1/2 = 1 - 1/2 N=2 : 1/2 + 1/4 = 1 - 1/4
N=3 : 1/2 + 1/4 + 1/8 = 1 - 1/8
...
N=n : 1/2 + 1/4 + ... + 1/(2^n) = 1 - (1/2)^n

And in the limit n goes to infinity, this gets arbitrarily close to 1. Which is what Achiles does when he is running according to Zeno's rules.

So mathematically, we deal with infinity using limits. We build finite systems, then we get our answer by taking the limit as some parameter goes to infinity. If that limit exists and is well behaved, then we can proceed to use that as the result.

The immovable-irresistable problem arises from a classic poser based on non-commuting limits.

Think about Achiles being able to run arbitrarily fast. And let the race track he must run be arbitrarily long. The time it takes him to run the track is just this.

T = D / S

Time equals distance over speed. Suppose we let him run at inifinite speed. Then the time it takes him to run goes to zero. If we take that limit first, then the distance has disappeared.

T = 0

And we can't get it back because there is no dependence on the distance. But suppose we did it the other way. Suppose we take the limit of the track distance being inifinite. Then the time it takes him to run it is also infinite.

T = infinite

And we have lost the speed he can run. So changing the order of these limits does not give the same answer. They do not commute. This refers back to the "well behaved" part of limits. When you have two or more limits in your problem, then if they do not commute they are not well behaved. (There are things you can do in such mathematical problems, but they don't fix our problem here.)

Which of the two things "immovable" and "irresistable" do we let go to inifinty first? Whichever we choose will dominate, causing the other to drop out of the result. And thus the limit is not well behaved.

Answer 3

There is no problem building a model that has both irresistible forces and immovable objects, so long as you also stipulate that they can never meet.

That can be done by restricting them to different parts of space, different times, or by defining 'objects' that the 'forces' don't apply to.

So, to take our universe as an example (and ignoring issues like all motion being relative and the effects of summing multiple forces that would render the question meaningless), it has already been noted that all forces are irresisible, in that when applied to any material object there will be a change in that object's momentum. But we can also define non-material 'objects' like a hole, a patch of empty space, a shadow, or the origin of my coordinate system. They occupy space, they have a definite position, but there is no matter there for a force to affect. Humans reify privatives like this all the time, and mentally model them as 'objects', provoking all sorts of philosophical debate as to where the boundaries lie.

Or to take another example, consider the centre of mass of everything that exists. Can it be moved? Newton's third law of motion says that for every action there is an equal and opposite reaction, any force applied by A on B results in an equal and opposite force by B on A, and while both A and B separately are moved, the centre of mass of the entire system is not. By conservation of momentum, it cannot be moved, because all forces are internal - there is no external vantage point from which an external force can be applied to the whole universe. The entire universe is thus an immovable object, containing numerous irresistible forces between all its parts.

As usual, it all comes down to how you define the words.

Answer 4

To get the contradiction, we need to build in "the irresistible force acts on all objects," or else the contradiction does not necessarily go through. "A force that acts on all objects," and, "An object that is not acted by any force, irresistible or not," are straightforwardly opposed and can only be combined with an exception clause, "There is a force that acts on all objects except for objects that are not acted on by any force."

To illustrate: suppose we say, "There is a tree taller than all skyscrapers," and, "There is a skyscraper taller than all trees." To fit these together, we have to say something like, "There are a tree and a skyscraper taller than all other trees and skyscrapers, except for each other." So presumably the supertree and the superskyscraper are equal.

Or consider, "There is a tree taller than all other trees besides the trees it isn't taller than," which is tautological.

Answer 5

It's not a question of inconsistency, it's a question of validity.

Physics is a mathematical model based on certain axioms. The aim of this model is to provide predictions about the real word. "What will happen if I throw a stone? Where will it land?" - these are questions about the future. Physics will then use a certain framework of definitions (mass, force, acceleration, rigid object...) and axioms (such as Newton's laws) to give you some numbers and curves; and then if you throw the stone corresponding to this model, you will know what happens before it actually happens.

Every model has a certain range of validity. If you give it parameters outside of this range, it will fail to give you predictions. For example, the singularity in the center of a black hole is a point outside of the mathematical validity of the Einstein equation; therefore, general relativity cannot predict what happens in that point, since it cannot be applied to that point.

When you are talking about an axiomatical immovable object and an axiomatical unstoppable force, you include two limit cases in the theory:

• Irresistible Force: F = lim(F --> infinity) - an infinite force.
• Immovable Object: m = lim(m --> infinity) - a rigid object with infinite (inertial) mass.

Now, what happens when the two meet each other? Your theory will certainly involve an axiom regarding this. In fact, since you're already talking about forces, you implicitly decide to use the Newtonian axioms (since they define forces); so we should stick to Newton's laws. Within this model, a certain parameter named acceleration (a) is the aim of prediction (and derived parameters such as speed etc., all of secondary interest for us now); and the axiom called Newton II gives the relationship between this and our two previously defined parameters, the mass m and the force F:

• a = F / m

But what does this model do when both F and m are infinite? Well, we get

• a = lim(F -> infinity) / lim(m -> infinity)

But calculus - the mathematical framework this model uses - tells us that this quotient can be anything! As such, "prediction" we get ist a can be literally anything. Our model is unable to make predictions about the movement of the object!

So what does this mean? Is our model broken?

No - it works as it should. It has a rigorous mathematical background, and its predictions work for almost all cases.

But this specific case is an edge case where this theory is simply no longer valid. Such cases, where we end up with unstoppable forces acting on immovable objects, are where the theory breaks down - we simply can't get useful answers by using it. And to overcome this issue, one must find a new theory that includes the case in its validity range.

Answer 6

It's definitely formally inconsistent, because either only one is true, or both have incompatible definitions that exclude each other.

Expanding out "irresistable force", you end up with "something that no barrier is physically capable of stopping" and expanding out "immovable object" means "something the position of which no force is capable of overcoming". In other words, an immovable object can't exist in a world where irresistable force is possible.

If we have the first, then the second is excluded from possibilities; likewise, with the second, it excludes the first from possibilities. A world where an immovable object is possible is a world where an irresistable force is impossible.

Which means that this an illogical statement. Wittgenstein would be yelling at you for saying it (or throwing chairs or whatever that brilliant/deeply-mad Austrian did for fun).

Although in fairness to the statement, it's not intended as a formal statement, and thus it's a lot less cleverly worded than the classic "God making an object he cant move" paradox that's not necessarily so easily resolved due to the muddling presence of the infinite astronaut.

Practically, an immovable object IS impossible (physics just doesn't work that way), and an irresistable force is probably impossible (though my physics knowledge is insufficient to prove that to myself and thus yourself). Although PERHAPS "escaping a black hole" might well be an interesting example of at least one of these axioms. Perhaps reformatted conceptually, being inside the Schwarzschild limit means you really do have an infinite "barrier" to butt your head against, which means, following the exclusion principle outlined earlier, you probably can't have infinite force.

Answer 7

... show that this system of axioms is in fact inconsistent because it is impossible to find a model that satisfies the axioms?

No: there is no problem with the model (the metaphysical perspective): the model is formally (although not numerically) consistent:

1) F = ma
2) F = ∞ 
3) m = ∞
Ergo
4) a = ∞/∞

Which means that the acceleration produced in the system is ∞/∞. Then, the problem of the model reduces to interpretation, which is not the same as its possibility (the model is possible, that value, which we'll perhaps never understand, is simply what the model predicts). The model is simple and there's no problem with it.

The problem is the application (the physical/empirical realization): it is impossible:

a) we don't know about physically infinite forces or infinite masses. In fact, the newtonian and relativist [1] interpretations are based on the empirical conception of bodies and forces. The problem is naive since empirically impossible objects are described with tools oriented to describe typical empiric objects.

b) any body subject to one of both conditions would easily dissipate, so in addition, we need an indestructible body and an invulnerable environment, which are not physically possible as well.

Notice that the Wikipedia page you refer to is a quite poor reference for this traditional problem, which can be formulated in multiple forms.

[1] https://uw.physics.wisc.edu/~himpsel/107/Lectures/Phy107Lect15.pdf