Sustainable Development, Science, and Logic. Part 1 - Logic.
Although we are in the 21st century, following several centuries of incredible science discoveries, and countless technological marvels, we still live in a world of false reasoning, obscurantism, and deceptions. In this series of posts I will try to explain the foundations of logic and science to show how these should be used in our everyday quest towards a more sustainable development and the truth in general.
In our quest for the truth, the whole truth, and nothing but the truth, we will start by making a "simple" logical demonstration to prove that the proposition "1 + 1 = 2" is true. Most people take this logical truth for granted while some people, confused with the definition of what a natural number is, believe that this proposition is either not true or at least not always true. We will hereafter prove that this proposition is, not only true, but always true, regardless of where we stand in the universe and even if there was no universe to discuss this proposition.
For those interested in the origins of this proof, it derives from the Peano Axioms postulated by Italian Mathematician Giuseppe Peano in the 19th century who was one of the first mathematicians to explore the use of logic to set beyond any doubt the foundation of all mathematical truths.
Before attempting to prove anything we must agree on definitions or postulates. The postulates are only required to allow communication between us, they do not need to prove anything.
The (hopefully minimum) set of postulates we need for this demonstration is:
P1: Let "0" be a "natural number"
P2: Let "1" be the natural number "succeeding" 0 in the "ordered" set of natural numbers:
"successor( 0 ) = 1" is true
P3: Let "2" be the natural number succeeding 1 in this set:
"successor( 1 ) = 2" is true
P4: Let "a" be a natural number, and let the "addition" of zero to "a" be equal to a:
"a + 0 = a" is true
P5: Let "b" be a natural number, and let the addition of ("a" with "the successor" of "b") equal ("the successor" of "a" added to "b"):
"a + successor( b ) = successor( a + b )" is true
Now that we have agreed on the meaning of what the natural numbers "0", "1", and "2" are, as well as succession, equality and what we mean by adding two natural numbers, we can show the demonstration. A logical demonstration is a set of logical steps using only postulates, or truths, to reach to a new truth. To be valid, each step of the demonstration must state the "truth" it relies on to be true:
P6: Let "a" be the natural number 1
P7: Let "b" be the natural number 0
Step 1: Using P5:
"a + successor( b ) = successor( a + b )" is true
Step 2: Replacing "a" and "b" by their values in P5:
"1 + successor( 0 ) = successor( 1 + 0 )" is true
Step 3: Using P2 ("successor( 0 ) = 1" is true):
"1 + 1 = successor( 1 + 0 )" is true
Step 4: Using P4 ("a + 0 = a" is true):
"1 + 1 = successor( 1 )" is true
Step 5: Using P3 ("successor( 1 ) = 2" is true):
"1 + 1 = 2" is true
This demonstration may sound over-developed to most people but this is necessary to build an absolute proof, beyond any doubt, that "1 + 1 = 2" is always true. Hopefully this is also the shortest proof (if you find a shorter, and valid, proof feel free to let me know).
For this truth to be true, we only needed to make postulates about the small subset of the three natural numbers 0, 1, and 2, as well as the notions of successor, addition and equality in this subset. So this truth is independent of any other predicates or truths. The independence of this truth from any other truths or predicates has far reaching consequences as we will now explore:
- Because none of the postulates or steps used in this demonstration assume a specific position in the universe, this truth remains true anywhere in the universe, it can be said to be independent from where we evaluate the validity of this truth in the universe.
- Because the postulates and steps used in this demonstration were independent from matter or energy, this truth can be said to be abstract, and remains true even in the absence of matter or energy. Therefore this remains true beyond our own universe and beyond any universe.
- This truth would still be true even if there was nobody to prove. The proof does not predate the truth. This truth was therefore true before its proof existed.
- This truth was also true before any human ever existed and before any universe ever existed and will continue be true after any universe would cease to exist.
- The proof needs someone do discover it but the truth itself does not need anyone to be true. So this truth can be said to be universal, to have always existed, to be independent of anything or any creator. Only the proof may need a creator.
- Hence this truth did not need any god or creator to be true, only the proof might have needed a god or creator.
Some might argue that without a proof the truth is irrelevant. To the contrary, without the truth predating, the proof would be impossible to discover. A logical truth is just as an island waiting to be discovered. Before being discovered by a human being, the island exists, and other lifeforms can flourish on it. Likewise, logical truths have consequences in our universe before being discovered by us.
The truth "1 + 1 = 2" is only one of many universal logical truths. The essence is to understand the universality of these truths, and our ability to verify each one of these truth by the use of logical reasoning. These truths and the logical reasoning to prove them constitute the solid foundation over which all scientific investigation and truths are discovered.
Logic should not be reserved to mathematical or scientific investigation, it should also be the basis of any proper discussion, problem solving, or conflict resolution. Unfortunately, more often than not, newscasters, politicians, business people, judges, and ordinary people use no or bad reasoning, bad and incomplete data, and no scientific approach in everyday conversations resulting in countless errors and suffering. We therefore need, more than ever, to re-establish the proper use of logic against false reasoning, obscurantism, and deceptions.
In part 2, I will explore the scientific method to help understand the incredible power of science and why we should rely more on science than we currently do.
