Languages generally break down into related groups that share a common ancestor; Indo-European is the best-known and -studied example, and yes, there was probably an actual proto-Indo-European language spoken about six thousand years ago in Southern Ukraine that ultimately gave rise to Bengali and Greek and English and Italian. Classically such groups are created by using a standard word list from each language and showing phonological correspondence between words, using a rigorous set of sound transformations applied consistent across the languages. Computational methods have more recently been added to the repertoire, which includes ways of looking at the grammar and morphology of the languages. These appeal to me because these elements are more conservative, allowing better resolution over greater times. It's pretty easy for individual words in languages to get replaced. But it's harder for elements of morphosyntax to change. They all have to work together in a much more rule-bound way, so it's like changing a single spark plug or belt in a car engine.
With respect to developing hypotheses about language descent and relatedness, linguistics are typically either lumpers or splitters. An argument for the lumper approach is that there's more value and more information in proposing a genetic taxon based on certain characteristics and investigating the hypothesis, than just putting every single language in its own category. The Altaic hypothesis is that there was a single language that gave rise to Turkish, Mongolian, Tungusic (northern China/southeastern Sibera), Korean, and Japanese languages. Including those last two has always been controversial. More recently even the reality of the Turkic-Mongolic-Tungusic core grouping has come under attack. For technical reasons of morphosyntax I won't go into here, I have always been quite sympathetic to the broader Altaic grouping.
Are there other media for comparative methods? Genetic studies have been an additional tool for understanding prehistory, and of course genes and language don't always covary: these studies have demonstrated that populations interbreed and learn new languages. We can also look at cultural practices, which may be more prone to drift than genes or language, but still provide intriguing signals; for instance, similarities in religion or superstition, which may also be more conservative than other elements. For instance, regarding the broad Altaic hypothesis (including Japan and Korea) you might notice that the cultural practice of steam or communal baths exists in Japan, Korea, and across the Turkic countriess from Kazakhstan to Turkey, but not in intervening countries to nearly the same degree (China, Siberia, India, Iran.)
An even more curious similarity is the superstitions that all these places have about whistling, which is also not shared by intervening cultures. In Japan, there are shinto spirits from the native pre-Buddhist Japanese religion called tengu, which are wind spirits. Also in Japan, whistling inside a house is considered to produce bad fortune, in that it will draw snakes into the house, or cause tengu to kidnap you. In Korea, whistling at night will bring spirits and snakes. (No word on whether fan death is related to this.) In Mongolia, you're not supposed to whistle inside a ger (yurt) because it will bring a windstorm. Tengri is the Mongolian sky (and wind) god. And you'll find the following in the "Respect" section of the Wikitravel entry for Kazakhstan: "Whistling inside a house is unacceptable in almost all Kazakh homes. It is a very common superstition in Kazakhstan that whistling inside will make the owner of the house poor." I wrote about this before but I keep stumbling across whistling superstitions in Altaic countries without even looking for them. While not rigorously part of the linguistic comparative method, this cultural comparison should increase our confidence in the broad Altaic hypothesis, i.e. that Turkic, Mongolian, Korean and Japanese populations are descended from a common ancestral population.
23 hours ago