What makes a definition a good one?

Diogenes of Sinope

Diogenes was a philosopher who expressed poverty as a virtue. He lived outside in a barrel and probably ate mud. He was known for being eccentric. He would often carry a lantern out in broad daylight and say that he was “searching for an honest man”. Plato, in his marble academy declared that “man” is a simply a featherless biped. Diogenes plucked a chicken and brought it to class and declared “Behold! A man!”. Plato later extended his definition to a featherless biped with fingernails.

A Good definition

I guess after establishing agriculture, humans decided to worry about meta problems, like what actually are humans? What is a man? Aristotle defined a man as a social animal. Plato, originally did a taxonomy. He said that there are bipeds and quadripeds, and among the bipeds, there are those with feathers, and those without. It follows then that humans are featherless bipeds. Why then, when, diogenes presented his gross, probably dead animal, did Plato not accept it? It is because his definition was wrong and useless. He doesn’t even use his own definition. Plato truly knows what a man is. Deep inside himself somewhere is an ideal of a man. On given input, he compares the input to this ideal, and then decides if it is a man on not. He is a glorified distinguisher. His “featherless biped” definition is just a failure of him trying to characterize his internal definition. His second definition, a “featherless biped with fingernails” will also fail, because I could present to him, a shaved gorilla. Then maybe he extends his definition to a featherless biped with fingernails capable of speech. Then I could attach a parrot to my shaved gorilla, and then he extends his definition to be all previous, plus having to be of a single brain. I think we could play this game forever.

Machine Learning

My motivation for that assertion actually comes from machine learning. Say you want an algorithm to detect if it has been given a picture of a bird. This is not something you can check arbitrarily, so you create a machine learning algorithm where you feed it millions of pictures of birds and it can eventually tell you if a picture probably contains a bird. The algorithm doesn’t actually truly know what a bird is. It has just generated hundreds of thousands of arbitrary conditions of which it checks the picture of the bird against. A machine learned program is just a bunch of if-statements. And its never totally accurate, just really approximate. A good definition should have no false positives or negatives. If Plato tries to come up with an arbitrary definition of man, I am certain I could beat it.

What is a Man?

I ask again, what is a man? What is a definition of a man? We argued we have an internal definition, that we cannot approximate arbitraily. To Plato, a man is then “whatever Plato says a man is”. This may be unsatisfying, but for a lot of things it turns out this is actually just fine. Take for example the discussion around the question “Are machines capable of thought?” This is sort of an extension of the previous question “What is a man?” because among all the unique characteristics of humans, our intellect stands out the most. The question of “Can machines think” has been reasoned about for centuries. Eventually Alan Turing gave the most concise definition if a machine is capable of thought with the Turing test. Essentially, an interrogator is able to ask certain questions to a human, and to a machine, and only using this is able to distinguish which party was the machine and which one was the human. This perfectly captures the internal definition of Plato. What is the the difference between a man, and something indistinguishable from a man? Nothing!

Cryptographic Definitions

In crypto, we quickly realized that arbitrary definitions are useless. For example take an encryption scheme and consider the definition “An encryption scheme is secure if the plaintext cannot be revealed from the ciphertext.” This seems fine, but then, like a plucked chicken, we can create a counterexample of an encryption scheme that always reveals the last bit of the plaintext. If it were a truly secure encryption scheme, then this should be impossible. This single bit might be all you need to infer the whole message. Our remedy for this is the IND-CCA experiments, where we test and adversaries ability to distinguish between any two ciphertexts given some computational power.

In secure computation we interact with this indistinguish idea more directly. We have two worlds, each with an equal set of players. In the ideal world, we have a trusted third party, who the players give all their inputs to and trust to perform the computation for them. This is in a way, really “ideal”. If there is some attack in the ideal world, then we really are helpless against it. The second world is the real world, where players have no party, but perform encryptions and message transmissions between players. We represent these two worlds as probability ensembles and say that our protocol in the real world is secure if these two probability ensembles is computationally indistinguishable (hence, “equal”).

Conclusion

Does this mean that all arbitrary definitions are bad? No. We have shown for some complex questions, a definition by indistinguishability is inherent, and one that is arbitrary will always fail. But some questions are simple and can be answered by simple arbitrary definitions. Take for example the defintion of an equivalence relation. It has to be transitive, reflexive, and symmetric. These turn out to be arbitrary conditions and seem sufficient. I can’t find anything that satisfies a correct equivalence relation, but in spirit, is “not equal”.

Things like intelligence, security, life, these are human constructs. Nature does not care if a system is “secure”, nor if something is alive or intelligent. These are things that cannot be defined arbitrarily, but have internal definitions. Maybe this is an easy way to define if something is a human construct or not. Can it be defined arbitrarily from some axiomatic description? If no, then it is a human construct. With this new definition, it begs the question, are human constructs a human construct? Lets just leave it at that.