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I get stuck in understanding the way the author phrases his thoughts in this blog:

We’ve already bet the kitchen sink on linear algebra and differential functions, we might as well just go all in and bet the farm on complex analysis.

I can grasp the point of the sentence but am not familiar with the analogy. I conjecture he means mainly that we can just take a plunge into unknown waters(complex analysis) since we already have benefited from the less powerful linear algebra and differential functions just for real numbers(real analysis).

Am I wrong? And how can I comprehend the "bet the kitchen sink on sth" and "bet the farm on sth else"?

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"The kitchen sink" metaphorically means "everything" -- actually, it's a truncation of "everything but the kitchen sink" (as in "He puts everything but the kitchen sink in his famous stew").

It's not really idiomatic to say "bet the kitchen sink". The idea of "everything but the kitchen sink" is you have some ridiculous collection of odds-and-ends, not anything of great value.

"Bet the farm" means to bet essentially everything you have. I imagine for the average small farmer, the farm does represent his only asset.

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Let me add to @Malvolio's answer

According to Cambridge Dictionary

everything but/except the kitchen sink

a much larger number of things than is necessary

This site (knowyourphrase.com) offers a lightly different interpretation

Including nearly everything possible

Your phrase doesn't use this exact idiom. If you have already bet nearly everything – all but the kitchen sink – there is not much difference with betting everything – the whole farm. I think that the kitchen sink in your context represents only a part of your "money" and the farm is the all-in, all you can bet.

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