When Lies Become True - Conjectures
Visualizing Daily Life Conjectures: Enhancing Understanding Through Imaginative Art.
What is Conjecture?
According to Wikipedia(January, 2022), a proposition that is suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has been found. So how can we believe something to be true without proof of it being true? The answer is conjecture via using past experiences to help predict future.
The prior patterns allows us to guess future outcomes to the best of our ability. That is what a conjecture is!
Famous Mathematical Conjectures
The term ‘conjecture’ has a lot do with mathematics, the ‘queen’ of the sciences, so it is appropriate to list a few of the more known conjectures in that subject.
The ‘ABC’ conjecture
Also known as the ‘Oesterlé–Masser conjecture’ .
https://mathworld.wolfram.com/abcConjecture.html
It relates the product of the coprimes of three numbers to the sum or max of them. In most cases, it is found that the prior is less than the latter.
A proof was supposedly found by Japanese mathematician Shinichi Mochizuki but without strenuous critique from mathematicians across the world. A Few years after he released a staggering 600 page proof, his achievements gained some recognition from the work of fellow mathematicians attempting to validate his work.
‘Twin Primes’ Conjecture
The belief is that ‘there exists infinitely many twin primes’.
A prime number p such that p + 2 is also a prime number. e.g. 11 and 13
The first few twin primes include:
(3,5)
(5,7)
(11, 13)
and so on
The truth value for this proposition is unknown still.
Infinite ‘Perfect’ Numbers
A perfect number is a positive integer that is equal to the sum of its positive divisors (including 1, excluding the number itself).
The first perfect number is 6
The divisors include 1, 2, and 3 which sum to 6 hence its ‘perfect’ status.
There are only 51 known perfect numbers and there are no known odd perfect numbers at the time of writing
Now that you have seen a few examples and maybe tried a few examples for yourself its hard to see why some of these conjectures have no real proof. As mathematicians we want to know the truth with solid reasoning an logic and without it we can’t make a decision on its truth value yet.
How Do We Conjecture?
Conjecture in daily life is more common than you think. For example, ‘if I go to university, then I will get a good job’. That is the consensus surrounding university but what exactly is a good job?, is the type of university considered? What if you don’t attend university all? Conjecture is sometimes not as clearly used in these cases.
If someone does not go to university it is fair to believe that if they did, they would get a ‘good’ job. There is sufficient evidence to believe so. However, it is just as likely and there is also sufficient evidence to believe that same person who didn’t attend university could get a ‘good’ job. Both cases have sufficient evidence to be true despite there actually being none according to how truth tables work.
Truth table for university(‘U’)-good job(‘G’) proposition
The first case is true as both U and G are true, the condition holds.
The second is false because G is false, U is meant is meant to imply Q.
The last two cases are significant because U is false in both cases but both evaluate to true. We should not confuse implication with multidirectional cause and effect. P as a hypothesis was not met so whether Q is true or false is irrelevant. The proposition is true in that case. This is the way to think mathematically but it sounds reasonable to employ in our daily lives.
If you didn’t go to university you could both get a good job and not which is not as trivial as it sounds. Both are a possibility.
Do We Conjecture in Our Daily Lives?
think the average human being will conjecture most decisions that they make. For instance, when it comes to meditation, most don’t test or collect results whether meditation worked for them. For most it’s guess work and there is a range for which they consider it useful. It can either be like “wow that was actually good” or “wow I didn’t feel anything”. That thought will decide the success of a conjecture in that case. It creates a statistic for someone to use as part of their conjecture. They assume that can be that case for themselves. It is an endless chain!
This is a similar case in sciences and mathematics as they favour hypothesis testing before they classify something as true. A conjecture becoming truth through chains of true statements from conjectures.
Business decisions can also require hypothesis testing. Think about data science and deep learning. The data comes first then after, analysis of data, predictive model building, validation of the model comes next. Finally, a general conclusion on a trend or phenomenon can be made.
A Conclusion?
We conjecture in maths because there is not enough evidence to give it’s definite truth value. However, many conjectures have a lot of truth to them but are just awaiting the missing piece to the puzzle.
