Difference between Buddhism and Aristotelian "middle-way"? This thought occurred to me after reading the following question and top answer:
How to refute the idea that Buddhism might be actually &quot;too extreme&quot;?

The questioner thinks the logical conclusion of Buddhism is to abandon all worldly goods, including the family, resulting in the elimination of the human race.


  Let's say if this is the appropriate thing to do, and everybody in the
  world does that, then the world, in 100 years or 200 years, will not
  have human beings any more (because if no marriage and no children,
  then there will be no new life on earth, while the existing people
  pass away). I can't say that it will be "good" if on earth, there is
  no more human beings, but lions, wolves, foxes, rabbits, and other
  animals remain on earth.


The answerer stated this is taking Buddhism to the extreme, and rather Buddhism is to avoid extremes and to take the middle path.


  Did Buddha teach "self-sacrifice"? No, he taught middle-way: by
  looking after ourselves we look after others, by looking after others
  we look after ourselves.


However, this is exactly what Aristotle said in his definition of virtue.  He defines virtue as the mean between extremes.

https://en.wikipedia.org/wiki/Golden_mean_(philosophy)#Aristotle


  His constant phrase is, "… is the Middle state between …". His
  psychology of the soul and its virtues is based on the golden mean
  between the extremes


Is there any essential difference between Aristotle's definition of virtue and what Buddhism teaches?  It seems there is only a difference in details, how specifically Buddhism teaches to seek the middle way, but the overarching principle is the same.  Or are there extremes in Buddhism's teaching?
However, this is exactly what Aristotle said in his definition of virtue.


Yes the ancient Greeks did have "nothing in excess" as part of their cultural heritage.

Buddhists might agree with some of the other 'wise' sayings on the page linked above, especially for example, "You should not desire the impossible".

But the ethics are not the same. Greek ethics are typically about being a good citizen, for example:


A "middle way" between cowardice and recklessness, so, being prudent and brave as a soldier
Being a just (justice) member of the jury
Being a political hero; etc.


The Greeks are into aesthetics too: they talk about "the beautiful".


  Is there any essential difference between Aristotle's definition of virtue and what Buddhism teaches?


You would do yourself a disservice to assume that, having read Aristotle, you have a handle on Buddhist Dhamma too.

Some essential differences include that they don't have the same goal, nor the same method/mechanism, not at all the same logic, nor the same experimental proof, nor the same practitioners.


  Or are there extremes in Buddhism's teaching?


Yes, I think there are many; so many that I cannot hope to pick "the best" example, so let me instead pick one example more-or-less at random.

The Itivuttaka starts with, "This was said by the Blessed One, said by the Arahant, so I have heard: "Abandon one quality, monks, and I guarantee you non-return. Which one quality? Abandon greed as the one quality, and I guarantee you non-return.""

In several ways, this is an "extreme":


It's said by "the Blessed One"
It's "abandon" one quality, not "see if you can't try to live without it quite a lot  of the time"
It's "I guarantee you", not "I'm pretty sure"
It's "greed", not "thinking about it"
It's "non-return", not "not too much of a return"


I'm not sure of the definition of the "middle way" you quoted (not sure whether it's a good definition: maybe you're quoting out of context).

Some more canonical definitions of the Buddhist "middle way" are as follows:

Dhammacakkappavattana Sutta: Setting the Wheel of Dhamma in Motion:


  "There are these two extremes that are not to be indulged in by one who has gone forth. Which two? That which is devoted to sensual pleasure with reference to sensual objects: base, vulgar, common, ignoble, unprofitable; and that which is devoted to self-affliction: painful, ignoble, unprofitable. Avoiding both of these extremes, the middle way realized by the Tathagata — producing vision, producing knowledge — leads to calm, to direct knowledge, to self-awakening, to Unbinding.
  
  "And what is the middle way realized by the Tathagata that — producing vision, producing knowledge — leads to calm, to direct knowledge, to self-awakening, to Unbinding? Precisely this Noble Eightfold Path: right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration. This is the middle way realized by the Tathagata that — producing vision, producing knowledge — leads to calm, to direct knowledge, to self-awakening, to Unbinding.


Cutting Off the Causes of Suffering:


  The Buddha calls this path the middle way (majjhima patipada). It is the middle way because it steers clear of two extremes, two misguided attempts to gain release from suffering. One is the extreme of indulgence in sense pleasures, the attempt to extinguish dissatisfaction by gratifying desire. This approach gives pleasure, but the enjoyment won is gross, transitory, and devoid of deep contentment. The Buddha recognized that sensual desire can exercise a tight grip over the minds of human beings, and he was keenly aware of how ardently attached people become to the pleasures of the senses. But he also knew that this pleasure is far inferior to the happiness that arises from renunciation, and therefore he repeatedly taught that the way to the Ultimate eventually requires the relinquishment of sensual desire. Thus the Buddha describes the indulgence in sense pleasures as "low, common, worldly, ignoble, not leading to the goal."
  
  The other extreme is the practice of self-mortification, the attempt to gain liberation by afflicting the body. This approach may stem from a genuine aspiration for deliverance, but it works within the compass of a wrong assumption that renders the energy expended barren of results. The error is taking the body to be the cause of bondage, when the real source of trouble lies in the mind — the mind obsessed by greed, aversion, and delusion. To rid the mind of these defilements the affliction of the body is not only useless but self-defeating, for it is the impairment of a necessary instrument. Thus the Buddha describes this second extreme as "painful, ignoble, not leading to the goal."[2]
  
  Aloof from these two extreme approaches is the Noble Eightfold Path, called the middle way, not in the sense that it effects a compromise between the extremes, but in the sense that it transcends them both by avoiding the errors that each involves. The path avoids the extreme of sense indulgence by its recognition of the futility of desire and its stress on renunciation. Desire and sensuality, far from being means to happiness, are springs of suffering to be abandoned as the requisite of deliverance. But the practice of renunciation does not entail the tormenting of the body. It consists in mental training, and for this the body must be fit, a sturdy support for the inward work. Thus the body is to be looked after well, kept in good health, while the mental faculties are trained to generate the liberating wisdom. That is the middle way, the Noble Eightfold Path, which "gives rise to vision, gives rise to knowledge, and leads to peace, to direct knowledge, to enlightenment, to Nibbana."


Glossary


  paṭipadā: Road, path, way; the means of reaching a goal or destination. The "Middle way" (majjhima-paṭipadā) taught by the Buddha; the path of practice described in the fourth noble truth (dukkhanirodhagāminī-paṭīpadā).


Wikipedia adds that "In Mahayana Buddhism, the Middle Way refers to the insight into emptiness that transcends opposite statements about existence." Wikipedia's footnote to this sentence references the Kaccayanagotta Sutta.