How do we know 1 + 1 = 2 ?

We step through the logic, one step at time, and at the end we agree it is a true statement. So at the end we know something is true because in the process we can remember, at each point we agreed with the logic and came to a conclusion.

So if there is a flaw in our process, we will only discover we were wrong, by finding it, and then confirming our result is now wrong. So perception is limited by the process and our memory of it. This shows how limited our interaction is with the world and how we take for granted so much, relying on processing systems to achieve our objectives, while not perceiving most of what is going on.

When we grasp the power of a true process though, we have something that is reliable enough to stake our lives on it.

So my conclusion is our confidence is in the process confirmed by our feelings after going through it, but we need to be careful we perform the process correctly.

So do we have more than the conviction ? But to me the conviction tells us everything is ok. And exams and learning show us, processes bring stability and success - and over-confidence without the correct process is just failure.

  • Had a professor as an undergrad that gave a proof of this: "Simple. If you have a circle in the sand and toss in a stone.... then toss in another stone, You get two stones!" In a less facetious tone, I think he meant that 2 can simply be defined as 1+1. Can rarely argue with a definition in mathematics. – user30473 May 9 '18 at 13:36
  • I am trying to make a simpler point about the process of conclusions. We know our conclusion is true by repeating the process. If you ignore the process, or forget it, you are still sure you are right because of the feeling. And one reaffirms this conviction by repeating the process, as many times as one needs. It would be nice if we could concurrently see truth, but we do not. – PeterJens May 9 '18 at 13:42
  • The pragmatic answer to knowledge is called truth correspondence theory. This involves using you famous senses to say something is so or not so. Much of what humans learn is through experience either your direct experience or through the stories of other human experiences. Truth is too vague of a word to use loosely as you do. Objective truths do not require any kind of agreement to be accepted. I would not use the word truth at all if humans have to agree for something to be so. You seem to be reporting human authorities have some thing to do with truth. I would say they are independent. – Logikal May 9 '18 at 13:44

How do we know that 1 + 1 = 2 ?

We prove it (as usually in mathematics) from relevant axioms.

See Peano axioms.

A mathematical proof is an intersubjective an surveiable process : it is written and recorded and we can review it many times.

See : Yuri Manin, A Course in Mathematical Logic for Mathematicians, (2ne ed.,2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

  • Most human knowledge comes through induction. That is, we use our famous senses that are sight, hearing, smell, taste & touch. When our senses match the real world this isa – Logikal May 9 '18 at 13:20
  • The matching of the senses with human experience is correspondence theory. Mathematics uses axioms which normal humans don't really do so unless they have training from education. Reasoning from axioms then must be taught. Aristotelian logic does not rely on axioms for instance & is closely related to how average Joe's reason in reality. – Logikal May 9 '18 at 13:38

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