What argument represents the thinking behind Protagoras's view that "man is the measure of all things" in Plato's Theaetetus?

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Plato's Theaetetus examines what is called the "homo-mensura" argument attributed to Protagoras. This argument is expressed in the saying, probably authentically Protagorean, that:

Man is the measure of all things, of those things that are, that they are, and of those that are not, that they are not.

Socrates and his interlocutors attempt to offer a fair understanding of this statement by re-interpreting it in various different ways. They begin by showing that naive interpretations can be absurd. For example, when Theaetetus has a fever after his war wounds become infected, his perceptions are not accurate. He may consider the room hot or cold based on his fever and may have hallucinations affecting his judgment. Thus it is not possible to consider this interpretation of Protagoras one that could be true. When Protagoras' head appears and accuses the interlocutors of unfair misinterpretation, they advance a more philosophical investigation of the nature of changing appearances, arguing that truth cannot be grounded in appearances, perceptions, or opinions for those constantly change. Instead, the truth must refer to unchanging noumena rather than changing phenomena.

A serious philosophical response within this framework, and one explored in Plato's Theaetetus, Sophist, and Cratylus, is that since the phenomena are fleeting and constantly changing, in fact, the changing and temporary perceptions of the individual are the only "measure" of these things. The problem lies not in man being the measure of phenomena but in the problematic ontological and epistemological status of the phenomena themselves.

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In Plato's Theaetetus, Socrates is undertaking a dialogue with Theaetetus, the Athenian mathematician, who is a devout follower of Protagoras of Abdera. Protagoras is the philosopher, called a Sophist, who claimed that man "is the measure of all things, of the existence of things that are, and of the non-existence of tings that are not." However, to claim that man is "the measure of all things" is to say that knowledge of truth rests on the opinions of man as individuals, which is also to argue that truth is relative. Truth is judged base on any particular man's perception of truth, and such a perception will not hold true for all men. Both Plato and Socrates were adamantly against the idea of relativism and instead argued for objective knowledge and truth. Hence, in Plato's dialogue, Socrates debunks Protagoras's argument by asking, why stop at saying that man is the "measure of all things"; why not say that a "pig or a dog-faced baboon ... is the measure of all things?" The purpose of his argument is to point out that, if we say truth is based on perception, or truth is relative, then we are also saying that truth is relative to all things, even baboons, not just man (Great Philosophers, "Protagoras: Plato's Critique"). However, despite the fact that in this dialogue both Plato and Socrates are vehemently arguing against the idea that man "is the measure of all things," or that truth is relative to each man's perception, there have been those that have held on to the philosophy of relativism throughout the ages, despite the fact that others have shown relativistic thought can "often lead to very implausible conclusions" (Standford Encyclopedia of Philosophy, "Relativism"). Nevertheless, a couple of arguments have been made that can still be rendered plausible.

One argument is stated from a linguistics perspective and of course has to do with differences in culture and language. Truths and beliefs must be expressed through words, but words can mean such different things per culture. Therefore, one culture will perceive one truth or idea being expressed, while another culture will perceive a completely different truth ("Relativism"). Author Joshua J. Mark of Ancient History Encyclopedia also cites another standard argument to substantiate relativism initially found in the Encyclopedia of Philosophy. This argument has to do with the relative perception of things like temperature. In short, the temperature to one person may seem cold, especially if that person is from a warmer climate, while the same temperature to another person may seem hot, especially if that person is used to colder climates. No one else would be able to dispute either person's claim that the temperature is hot or cold as "the measure of hotness or coldness is fairly obviously the individual person" (as cited in "Protagoras of Abdera: Of All Things Man is The Measure"). In other words, it can be said that we cannot argue the objective truth of temperature because temperature is experienced on a completely subjective level.

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