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Can something be objectively meaningful if meaning itself only exists inside human minds?

Is it possible, simply through the study of psychology and without knowledge of mathematics, to tell that 2+2=5 is wrong? Simply through the survey of the internal contents of the mind, can you make any judgement about what is true or false? You cannot tell whether affirming the consequent and denying the antecedent are fallacious forms of reasoning or not. For that, psychology has to turn to other sciences, and so it must necessarily presuppose others.

Or take Transcendental Phenomenology, which is the study of phenomena - the subject matter of this science. To arrive at this phenomena we have to abstract from real existence and concentrate on how things appear to consciousness. This epoche, as Husserl calls it, drives a wedge between meaning and reality, but this separation is presupposed within the science and not justified within it. The phenomenological reduction is, as a consequence selective; it knows what it negates and what it preserves. There is already a duality of meaning and reality built into it.

So, the answer to your question is no, in order to be true, a content must not solely exist in my mind but should also be true about external reality. Also, we cannot assume that these meaningful contents in the mind can be individuated without relation to external reality at all.

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