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Angst and the Empty Set

We can experience nothingness, but does it actually exist?

Suppose you open your handbag one day expecting to find your wallet there, but don’t. Do you literally see the absence of your wallet…By Leon Horsten

Suppose you open your handbag one day expecting to find your wallet there, but don’t. Do you literally see the absence of your wallet in your handbag? If you do, it means something important: Absences have a positive presence in your perception that you can grasp, independently from all ordinary things.

Anna Farennikova, a philosopher at the University of North Carolina at Chapel Hill, has argued exactly this position. The phenomenology of the experience of seeing something missing, she says, has “immediate, perceptual qualities” that encourage us to see absences as being as fundamental as presences. Even though our visual experiences of absences are triggered by frustrated expectations, it feels like we are literally seeing absences, rather than deducing them.

Another, much older school of philosophy also holds the more radical view that we are acquainted, not just with the absence of things, but with nothingness per se, through our experience of existential anxiety or angst. Called existential phenomenology, this school defines angst as an emotional state that, unlike fear, does not have a clear object. The philosopher Martin Heidegger maintained that when in angst, one is in contact with nothingness. In Heidegger’s view, we can experience not just the absence of things, but we can experience nothingness itself.

Science, too, seems to suggest that there exist many “nothings” that have a positive existence: the vacuum, the number zero, and the empty set all seem to be objects with positive qualities, which can be pointed to, isolated, and manipulated.


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But some of these arguments may not be as convincing as they appear. For example, under closer scrutiny, all of the “nothings” considered by science are actually something. Let us take the vacuum first. A vacuum is a region of space devoid of matter. But according to quantum mechanics, a vacuum contains quantum fluctuations and has non-zero energy. And in any case, if a vacuum is a region of space, it contains space. One might seek to reduce space to relations between concrete particulars, but that is a difficult project. So the vacuum is not nothing.

Concerning the number zero, is it really identical to nothing? It is tempting to reason as follows. Take any number x, and add nothing to it. Then you get the number x back. Now take x, and add the number zero to it. The result is the same. Therefore, the number zero is nothing. But this reasoning is fallacious. One could argue in a similar way that the number one is nothing: Take any number x, and do not multiply it by any number. Then you get the number x back. Now take x, and multiply it by one. The result is the same (x). Therefore one is nothing. No one would agree with the conclusion of that argument!

In set theory, the empty set, which is the set that contains no elements, is taken to be a fundamental building block of the mathematical universe. Even though it looks like nothing at first sight, this, too, is on closer inspection seen to be not nothing. The empty set is like an empty shopping bag. But unlike a shopping bag, the empty set is abstract. And just like an empty plastic shopping bag is not nothing, an abstract empty shopping bag is not nothing, either.

Science, then, does not in any straightforward way tell us that there are nothingnesses. According to the scientific picture of the world, absences do not seem to be fundamental building blocks of either the concrete (physical) world or of the abstract (mathematical) realm.

There is also a philosophical view about absence that is in agreement with this scientific worldview. It holds that absences can be reduced to presences, and that a fortiori there is no such thing as nothingness. This view was worked out in detail in mathematical logic in the early 20th century and defended forcefully by Bertrand Russell.

Today, this alternative view that absences of things can be reduced to presences of things is still the dominant view among philosophers. The chief reason for this is that as far as we can tell, the scientific worldview can do without absences of things as positive entities. Moreover, science certainly sees no need to attribute an irreducible existence or presence to nothingness.

But hold on. The term “nothing” can be correctly used in assertions, as in “I said nothing.” So expressions such as “nothing” are certainly meaningful. If “nothing” refers (to nothingness perhaps, or to an absence of an utterance…), then its meaning is clear. But how can the view that seeks to reduce absences to presences explain the meaningfulness of the term “nothing?”

Russell has a good reply to this challenge. According to Russell, the meaning of the term “nothing” is not mysterious at all. It can be spelled out using logical vocabulary only, with just the logical operators “for all,” “not,” and “equality”: A situation in which there is nothing is a situation in which the statement “for all objects x, x is not identical to x” is true.

If to give the meaning of an expression is to give the conditions under which it is satisfied, then it seems that we have hereby given the meaning of the term “nothing.” But the meaning of “nothing” has now been given purely by quantifying over things that do exist. So in some sense, the meaning of “nothing” is not self-standing but parasitic on quantification over objects and on negation.


All of this still leaves us with our original question: Are absences fundamental in perception? After all, there is a distinction between the hypothesis that absences are fundamental in nature and the idea that the seeing of absences is fundamental in the theory of perception. On this question, the jury is still out. Whatever the case may be, it is clear that reaching for your missing wallet involves more than meets the eye.


Leon Horsten is a philosopher of mathematics and a philosophical logician at the University of Bristol.

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