Everyone is Religious
So, what do you know about the man on the left? More importantly, how do you know it? Teachers may have taught you about his impact on the World. You may have observed drawings of him, monuments to him, places and people named after him, personal accounts of his deeds and existence. Realistically, any information we can know about him are recorded in books or online archives.
What about the guy on the right? Clearly his existence stands the same burden of proof, right? Maybe not. After all, Jesus could walk on water, turn water into wine and perform all sorts of miracles that automatically disqualify him from being a real historical figure.
Well, Washington allegedly never told a lie. How believable is that? By this line of reasoning Washington could never have existed, which is obviously silly. Jesus is ridiculed by modernity, while a slave owning, alleged Native American murdering, free mason is a hero. Jesus had an ethical code from above, while Washington wrote his own rules. The common denominator to believing in either is faith.
Here we see the deification of President Washington, resurrected in Heaven sitting among the angels. This painting in the form of a circumpunct spans the eye of the rotunda in the capital building, located appropriately in Washington D.C. Suffice to say, many people thought of him as a demigod including Mason Weems, the author who gave us the famous choppin' down the cherry tree tale. From a secular understanding, Jesus was born the son of God and Mary, by very definition a Demigod. To say Jesus never existed is hypocrisy. To say faith never mattered is denial.
What can we Know?
In many cases, faith is looked down upon as a basis for Truth. We are told only logic and reasoning could ever get us there. I'd like to explain (with logic and reasoning) why this is not the case.
The Limits of Logic
To begin, we must define at a high level what logical systems are. They are describers. Any mathematical system you think of must appeal outside of itself to absolute truth. They serve to describe truth in meaningful ways. Math is descriptive and science is explanatory. Science being downstream from math, simply explains causal relationships. Both these fields presuppose absolute meaning and require a foundational grounding. This is where the branches of metaphysics and epistemology in philosophy fit in. To say philosophy does not belong in science is a philosophy unto itself. Objectively, all mathematical and logical systems are incomplete, inconsistent, and undecidable.
A (Recent) History of Math
In 1874, Georg Cantor published a new paper which gave birth to Set Theory, a set being a well defined collection of items. He went on to describe the cardinalities of the sets of Natural and Real numbers. Respectively, one was countably infinite while the latter was uncountably infinite. By a diagonalization proof, we could list out at random the decimal expansions between 0 and 1 in a table. By adding one to each digit on the diagonal, we will construct a new number between 0 and 1. Therefore, we could never truly list out the real numbers, even on the domain from 0 to 1.
This was very controversial in the fields of Mathematics. Up until that time Euclid's Elements were the bread and butter of math. At the turn of the nineteenth century non-euclidian geometries were proposed by Gauss who disregarded the axiom that triangles can't have an angle sum of greater or less than 180 degrees. Mathematicians were beginning to look more intently at the foundations of the logical system we know as math.
A schism in the field of math gave way. The Intuitionists thought Cantor's work was nonsense. Math was a pure creation of the human mind. The infinities of Cantor were not real. The Formalists thought math could absolutely be compatible with set theory, the iconic leader being David Hilbert.
Hilbert was convinced a more rigorous form of proof based on set theory could solve all the lingering issues in math that were still not addressed. However, by 1901 Bertrand Russel pointed out a serious problem with set theory, mainly a paradox of self reference.
First, we could see that the set of all sets must contain itself. The set of all sets that contain themselves starts getting fishy when we look at its contrapositive. What is the set of all sets that don't contain themselves? Let's call this set S. If S doesn't contain itself, then it must contain itself. But if S contains itself, then it doesn't contain itself! This is the barber paradox in disguise. To resolve this, they restricted the definition of a set to exclude any self reference.
However, the problem of self reference did not suddenly vanish. Formal proof methods were devised to once and for all solve this problem. In Principia Mathematica, a book of notoriously dense and exhausting notation, it took 762 pages to prove 1 + 1 = 2. The fourth Volume was never published for obvious reasons. Needless to say, this type of mathematical reasoning is exact, making linguistic methods obsolete. Most importantly, it allows us to prove formal properties of the system itself.
The Big Three
Hilbert sought to prove 3 questions true. Is there a way to prove
every true statement? In other words, is Math complete? If it is we do
not need faith and we're done. The second being,
consistent meaning is it void of contradictions. If we could prove
A and NOT A then we can prove false statements true. The last one is,
math decidable?. That is, does there exist an algorithm which can
always determine if a statement follows from the axioms. Hilbert thought
the answer to all three questions was yes.
Kurt Gödel proved Hilbert wrong. A complete formal logical system is
impossible. His proof is as follows. Consider all of the basic operators
in math, such as +, -, =, <, parentheses, conditionals, negations,
existential and universal quantifiers etc. Each operator would be a
card with a Gödel number on the back starting at 0 up until infinity.
Numbers themselves would be represented in a slightly cumbersome way.
Zero would be given some value ad hoc, say 6. A new operator called
successor of labeled s would be combined with 0 to make
1,2,3 so on and so forth. For example
2 would be 3 cards:
Now we can start forming equations such as 0 = 0 for instance. 0 is 6 and = is 5. Laying the cards out we get 656. To construct a new card and its Gödel number we make the digits the exponents of prime numbers and multiply, so in this case we'd have 26 * 35 * 56 = 243,000,000. So 243 million is the Gödel number of 0 = 0.
What's powerful about this is we can write Gödel numbers for any statement you could imagine. We have an infinite deck of cards where any set of symbols in some sequence is encoded by a unique fingerprint. What's even more powerful is we can take the prime factorization of any Gödel number and work out the basic symbols it's comprised of. In this stack of cards there are true and false statements, thus there is a system of proofs.
To prove something is true, we construct axiom cards. The first being ~(sx=0) meaning the successor of any number is not zero which is true considering negative cards don't exist in this system (Gödel numbers can't be fractional). Substituting 0 for x, we get 1 does not equal 0. This proof also gets its own number. We take 2 (axiom's number) * 3 (number for 1 ≠ 0) which is a number 73 million digits long. Since things get large so quickly, Gödel just labels them as letters.
Gödel jumps through all these hoops to assess the card labeled,
There is no proof for the statement with Gödel number g. The
catch is this card's number is g. This statement is conveying
that the card itself is unprovable. There is no proof in our infinite
deck, if there was, then you have shown there is no proof. But if
there is a proof which shows there can't be a proof we've arrived at a
contradiction. This would mean mathematical systems
are inconsistent. On the other hand, if there is truly no proof for the
card in mind, then there exists true statements which cannot be proven.
What we get from this is Truth and Provability are not the same thing.
Hilbert was quite displeased but still was convinced Math is consistent. Then Gödel published his second Incompleteness Theorem which showed any consistent formal system of math cannot prove its own consistency. At this point we are left with faith. We must have faith that the system of logic and reasoning we use is consistent. We also see that the absence of a proof does not imply a claim is invalid. We have one last speculation: decidability.
This question was not solved until computers came around. Through his contributions to the field of Automata Theory, Alan Turing proved that some problems are undecidable. He called this the halting problem. This is because his Turing Machine would halt when the program ended. The problem itself is given any input of 1's and 0's can we determine if the TM halts. Obviously we can have infinite loops where the TM goes left and right forever. But is there an algorithm which could tell us right away if this happens? What if the program takes a billion years to terminate, how could we programmatically test this?
Let's say there were such a program called h. We give
h a program and some input, then it will simulate what will
happen and output a 1 or 0 for
respectively. We can add a wrapper for h. If h outputs
halts the new wrapper program will loop forever. Conversely if
the program h outputs
never halts, the new program halts. This means
our wrapper will now only have 1 output.
Let's call this new program h+. We export the code of h+ and pipe it into h+'s input slot and its program slot. Now, h is simulating what h+ would do given its own input. h has to determine the behavior of a machine that it itself is apart of. If h concludes h+ never halts then h+ halts. On the other hand, if h concludes h+ halts, then h+ never halts! This means h cannot exist.
An application of undecidability is trying to solve a rubik's cube in at most 20 steps. Many people can't solve them to begin with, but to be able to solve any rubik's cube in at most 20 steps is not possible by any human. To be clear any rubik's cube can be solved in at most 20 steps, but there is no single unifying algorithm which solves all 43 quintillion different states in under 20 steps. It's not called God's algorithm for nothing.
Hopefully, I have given you the reader some insight on why faith is
necessary. There can be no
Theory of Everything, because any
truth claim has to appeal outside of the system it belongs to.
Anyone who forms an argument is appealing to faith. Everyone is religious.
Truth is conceptual and absolute. Concepts require a mind. Human's did not make truth, nor could we ever completely know it. Therefore an absolute and eternal mind exists, and is responsible for creating meaning, providing a basis for truth. How can I prove this? I can't. But we can recall no logical system will ever be complete. What's rational to us is what's comprehensive to the human mind. There are true statements which we will never be able to understand. That's why we need faith.
What’s reality? I don’t know. When my bird was looking at my computer monitor I thought, ‘That bird has no idea what he’s looking at.’ And yet what does the bird do? Does he panic? No, he can’t really panic, he just does the best he can. Is he able to live in a world where he’s so ignorant? Well, he doesn’t really have a choice. The bird is okay even though he doesn’t understand the world. You’re that bird looking at the monitor, and you’re thinking to yourself, ‘I can figure this out.’ Maybe you have some bird ideas. Maybe that’s the best you can do.-Terry A. Davis