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Understanding Infinity and Infinite Regress


Visual by Akruthi Akula


The problem of origin is one that has remained unsolved since the advent of thought. Philosophers throughout history have tried to address the mystery of “the uncaused cause” and “the unmoved mover”. These mysteries are a result of attempts to prevent the problem of infinite regress, in which the search for justification is unending as the question of justification of every next/previous premise keeps arising.


To defend a theory, or prevent it from falling into the problem of infinite regress, an infinite regress argument is used. One example of the use of such an argument is in the argument for the existence of God through The Cosmological Argument given by Thomas Aquinas.



Epistemic Regress and Metaphysical Impossibility


In philosophy, there are two major forms that the infinite regress problem takes.

Epistemic Regress: Knowledge is a true, justified belief. In every chain of logical reasoning, each step depends on the previous one (its justification) for its existence. For the sake of understanding, let’s take a step T1 in a logical argument. T1 relies on T2 for its justification, and similarly, T2 relies on T3, which again would require a justification in the form of T4. T1 will never be supported adequately, because the needed series of support would be infinite. Thus the search for rational justification is neverending, as it seems that a starting/ending point can never be reached.


Metaphysical Impossibility: This can be explained by taking an ontological example. Things in the external world are taken to exist because of the parts that they are composed of. These things (parts) must also depend on their own parts for existence. This chain continues till infinity, which many thinkers like Thomas Aquinas claim to be impossible.


There are things in the world (composed of matter). Everything has an origin. Something cannot come from nothing. Yet, something has to come from nothing if there is to be an origin. The idea of an origin-less universe seems absurd, yet an explanation (without contradictions) to escape this has not been obtained so far. The problem of metaphysical impossibility is thought to be the most serious form of viciousness entailed by infinite regress.



Aristotle and Francisco Sanchez on Infinity and Infinite Regress


Aristotle

The Greeks viewed the concept of infinity as being synonymous with chaos. Apeiron (Greek for infinity) was a problem for them because it evaded their conception of everything existing in precise and measurable terms.

Aristotle defined infinity as being a potential state. He said that the existence of infinity was necessary, because he couldn’t conceive time as having a beginning or an ending. Nor could he conceive the same when it came to counting numbers, or of the universe. But, at the same time, he said that infinity coouldn’t possibly exist. To support this stand, he said that the existence of an infinite object is unimaginable, because by virtue of its infinite nature, it is undefinable, and hence could not be told apart from other objects. To conclude, Aristotle said that as neither of these two stands seems possible, infinity must both exist and not exist in different senses.


Francisco Sanchez

Sanchez took Aristotle’s argument further by giving an argument for skepticism based on infinite regress. Going against foundationalism, while supporting evidentialism, he pointed out the absurdity of the idea that knowledge rests on something other than knowledge. From this, he concluded that nothing could be reasonably believed. One of his popular quotes is - “Knowledge is fiction.”


The argument he gave is-

Premise 1: Knowledge cannot rest on assumption.

Premise 2: Knowledge requires justification

Premise 3: Knowledge is finite.

Conclusion: Nothing is known.


This is known as the semantic argument. He used it to convey that even though the meaning of knowledge is known, it cannot possibly be obtained. Therefore, we need to substitute the word knowledge with reasonable beliefs.



Is the IR Really Vicious?

An infinite regress argument is considered vicious if it leads to a “metaphysical impossibility”, if it causes its theory to become “implausible” and if it does not solve the problem underlying its origin. However, there are non-vicious or virtuous regress if they do not go against the theories that require them for eg. the truth regress: if the proposition "P" is true, then the proposition that "It is true that P" is also true and so on.


Infinite regresses that have a metaphysical impossibility are the most common vicious regress since by following the assumption that “actual infinities” are not possible the argument becomes contradictory. Such regress could be infinite magnitudes like infinite densities. Infinite regresses based on implausibility call for the denial of its actuality rather than possibility, for example, the argument of the existence of God suggests that God is the first cause. Other examples of this kind of viciousness involve human actions. The failure of a theory to explain a problem also leads to viciousness for eg. the informal fallacy of “begging the question” where the premises assume the truth of the conclusion. The famous expression of infinite regress turtles all the way down states that how mythologically earth rests on the back of a turtle which rests on the back of another turtle and so on. In this expression though there is metaphysical possibility since space can be infinite but still it is vicious because this argument is supposed to explain why the earth is not falling but rather than explaining that it simply states that there is another thing which is not falling. This expression is vicious as it does not explain “why nothing at all is falling”.


Positions to Avoid IR

The question arises whether the infinite regress is really vicious and philosophers who are infinitists state that it is not. Two main responses to avoid the regress are foundationalism and coherentism.


Foundationalism can be found in epistemology and it states that there is a first cause from which things arise but which does not itself arise in this manner. In epistemic theories, the foundationalist approach follows that there are two kinds of beliefs: “inferentially justified beliefs” which are dependent upon other justified beliefs and “non-inferentially justified beliefs” and the second are the foundation for all the other beliefs. Foundationalism is also found in metaphysics where it is considered that some things exist at a more fundamental level than others which are dependent on them allowing for no infinite regress.


Coherentism is the second response also found in epistemology which avoids the regress by considering things to be interconnected rather than being linear. This approach allows justification of beliefs based on their being “coherent” with each other.



Is the IR Problem Still Relevant?

The problem of the infinite regress has been a debate among philosophers through centuries and due to its applications in epistemology and metaphysics, it is at present as relevant as ever.



Food For Thought

There is an interesting question that helps explain the problem that infinity, as a concept, poses. It has no conclusive answer, as of yet. The question is - “If you turn a light bulb on and off an infinite number of times, does it end up on or off?”

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