Suppose a mathematician wants to find easier ways to calculate the square root of 4-digit numbers. He randomly picks 2025 as his starting point; the square root is 45. He suddenly makes an interesting discovery:
20 + 25 = 45
This makes him think that to calculate the root of 4-digit numbers, the number has to be separated in the middle into two different numbers, which will result in the root when added. He then randomly picks 3025 and 9801. Their square roots are 55 and 99, respectively, which corroborates his findings:
30 + 25 = 55
98 + 01 = 99
The mathematician is now confident that his theory is true, and it constitutes absolute knowledge for it has been proven by his observations. This scenario, found in Malba Tahan's The Man Who Counted, is the embodiment of philosopher Karl Popper's issue with empiricism and induction.
Empiricism argues that theories can be verified through evidence, closely connecting with induction–the process by which we draw conclusions from observations. Popper argues that induction is flawed, for it assumes future observations will align with past ones (as originally suggested by David Hume during the early 1700s), just like our mathematician assumed the rest of 4-digit numbers would align with the pattern found in his 3 observations. Similarly, he criticizes empiricism's claim that its evidence can verify knowledge, because there is always a possibility for contradicting evidence to emerge.
Is there any verifiable knowledge, then?
Popper holds the fallibilist belief that no knowledge is absolute. Thus, he considers that while empirical evidence cannot verify knowledge, there are two things it can actually do:
1. Help formulate conjectures (or theories/hypotheses)
2. Falsify theories (falsify: prove to be wrong)
This places an important emphasis on the testability of ideas. For knowledge to be scientific and valid, Popper claims it should be falsifiable (testable). Those theories that can stand the attempts at falsification are the ones that can be considered knowledge (at least until proven otherwise).
But, what does this mean for us?
Popper's beliefs about the provisional nature of knowedge should remind us that we are fallible, there is always a chance that we could be wrong. They highlight the importance of being skeptical, but also open to new ideas, keeping in mind the sheer irrationality of believing any of the perceptions we hold about the world is an absolute and indisputable reflection of reality.