Logical cognition is a process of using logical reasoning to draw conclusions from premises. However, the correctness of the conclusions is not guaranteed, because the logic may not be self-consistent or the parameters may not be complete.
Logical cognition, such as mathematics, is much better than thinking cognition. Here's another example: Imagine a typically reserved person suddenly going on a rampage with a gun on the street. It would be incorrect to claim this person has a good temper, wouldn't it? Similar characters are often portrayed in online videos, appearing mild and sincere but retaliating without apology if provoked. They are genuinely dangerous! Such individuals do exist in real life. Can we genuinely say they have a good temper? We might perceive them as having a good temper simply because they do not talk much, but beneath the facade, they might harbor a lot of anger, and when it erupts, it can be profoundly alarming.
The logic represented by this example is called common-sense logic, and there is another type known as mathematical logic, as seen in mathematics. Mathematics is quite accurate, but common-sense logic can often be wrong because the way we look at things is often one-sided, only seeing a part of the big picture within a limited period of time. For example, if you see someone who is very gentle, you might assume he has always been like that. However, perhaps ten years ago, he was a very irritable person, and later he was beaten up by someone, and since then he has become gentle.
We often view things with strong temporal, regional, cultural, and one-sided biases, which often lead to common-sense logic being wrong. In contrast, mathematical logic is relatively more rigorous. However, it inevitably grapples with some inherent issues in logic itself, such as the three crises in mathematics— the Pythagorean paradox, the Berkeley paradox, and the Russell paradox.
Because mathematical logic cannot be theoretically self-consistent, it can also have problems. Although mathematics is incredibly powerful within certain areas, it does not guarantee absolute truth.
For instance, when you go to buy vegetables, it is a very straightforward process. If the vegetables cost 8 dollars per kilo, and you need 3 kilos, 3 times 8 is 24 dollars, which can be calculated immediately. But mathematics is not that simple. For example, √2≈1.414…… It is an irrational number, an infinitely repeating decimal. Such a number cannot be found in reality; it can only exist in concept, posing a logical contradiction.
Another example is the circle ratio π. Can you find a rope exactly as long as π? Absolutely not! The decimal representation 3.14159... goes on indefinitely.
Calculus faces a similar question: "Can calculus be completely divided?" To address this, it involves the axiom of Neither One Nor Many of Madhyamaka; if calculus is divided completely, it ceases to exist as a subject of study. If it is not divided completely, then to what degree should it be divided?
This means that mathematics can be applied, but it cannot be absolutely self-consistent in logic. For example, the Barber Paradox, which I will not talk about. For the fellow practitioners who like to listen have heard it many times. For those who do not like it, even after hearing it so many times, still act like they have never heard it, and they get annoyed every time they hear it. The Barber Paradox is a crisis of sets. It has not been solved, neither is it logically consistent. Therefore, the correctness of conclusions drawn from logical cognition is also indeterminate.
I deny the claim that mathematics and science are absolute truths. Science is just a collection of many so-called empirical cognitions that cannot be absolutely verified. To deny that mathematics is the foundation of science is to deny all of the existing cognitive methods of mankind. While mathematical logic excels in its practical application, claiming it as an absolute truth may not be entirely accurate.
Excerpted from:Cognition and Expression Part One
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