“ It is no crime to be ignorant of economics, which is, after all, a specialized discipline and one that most people consider to be a ‘dismal science.’ But it is totally irresponsible to have a loud and vociferous opinion on economic subjects while remaining in this state of ignorance.
“ And all this madness, all this rage, all this flaming death of our civilization and our hopes, has been brought about because a set of official gentlemen, living luxurious lives, mostly stupid, and all without imagination or heart, have chosen that it should occur rather than that any one of them should suffer some infinitesimal rebuff to his country’s pride.
Bertrand Russell in “The Autobiography of Bertrand Russell (via philphys)
The Problem of Induction
Let’s say you’ve only seen white geese throughout your whole life. With this in mind, someone approaches you and asks you to describe the next goose you will see. Thanks to your past experiences, you cannot help but mention that it will be white.
Essentially, to conclude that all objects contain a property based on the observation that this property is contained within a few objects is called induction.
This tends to be a successful at times. When asked if the sun will rise tomorrow, one can merely cite all the past times that the sun has risen. Or when attempting to apply a scientific law, we can be sure that we will succeed based on the success of all the past instances of success. In mathematics, induction is powerful enough to allow mathematicians to use it to define and prove properties of numbers and mathematical objects.
For instance, it’s not difficult to prove that the square of a positive integer is the sum of the odd postive integers of that size:
- Base Case: 1 = 1^2, 1 + 3 = 2^2, 1 + 3 + 5 = 3^2
- Inductive Step: Assume that this is true for k.
We want to show that this is true for (k+1),
ie. 1 + 3 + … + (2k-1) = k^2 is true,
so we add (2(k+1)-1) to both sides:
1 + 3 + … + (2k-1) + (2(k+1)-1) = k^2 + 2k + 1
which is equal to:
1 + 3 + … + (2k-1) + (2k+1) = (k+1)^2
Beautifully simple, right?
Induction has proven itself to be useful in this way. However, induction suffers from a logical inconsistency. See, induction functions so well in mathematics because the objects that are being studied have a rigid definition. No mathematician will present confusion at the notion of a positive integer.
Yet, consider the example given before. Is it correct to conclude that all geese are white based on having seen just a few? Definitely not, as “white” would then be part of the definition of “goose.” And certainly, instances of non-white geese have been observed. Similarly, it is not in the definition of “sun” that we will observe it rise everyday. In fact, it is possible that something may occur to impede the sun from rising. That we cannot check this to be true enlightens on the nature of this logical inconsistency. With mathematics, there was the luxury of checking any instance I wanted. But in reality, I cannot check every instance of the run rising to make sure that I am right in concluding that the sun will rise tomorrow. The same applies to any scientific law, as no one can check every instance of gravity to make sure that the laws governing gravity will always hold. To suppose that we do have knowledge would mean to conclude what we want to prove, which itself is a logical fallacy.
So, when working in mathematics, it’s perfectly acceptable to make assumptions about some object that is aapproached. In reality, doing such a thing can easily lead us astray.
aand heres to -hic- not fukcin thsi yearr up asb ad as th lsat one„„
I lied awake.
The consequences of that were typical. It meant I was probably tossing and turning, unable to find an optimal sleeping position. It meant that I was going to wake up feeling tired. It meant that I’d beging falling asleep in the middle of doing work tomorrow. (Or today, since it’ll technically be today) It meant that I was probably mentally fixated on something. It probably meant something was bothering me.
Yes, that seems right.
So I was faced with this dilemma: I could continue to attempt sleeping or I could remain thinking. I had been doing both, and that wasn’t working.
I looked around. My eyes had yet to adjust to the darkness, so I was blind to my surroundings. I could hear, but I was surprised by the quietude. I suddenly started imagining that I was encapsulated by a dark fog that swallowed all light and sound.
“Hello,” I whispered.
As quickly as I had made the sound, it disappeared. I failed to dissipate the quietude. That was disappointing. So I tried again, snapping my fingers. I heard the short snap under my sheets, but it was only a temporary break from the quiet.
Wait, why am I doing this?
I didn’t want to wake anyone up with my loud thoughts. Or perhaps I did. I sat up, feeling the cold carpet under my toes. I stood up. Surprisingly, the floor creaked with each step I made. I hadn’t noticed that before. It inspired me to tip toe out of the room, taking my time to open and close the door. I walked though the hallway and into the living room. There was a rectangle of yellow light on the floor, whose origin was the windows that faced the street. I sat down on a sofa and inspected this phenomenon, focusing on it and closing my eyes. I tried for a third option; clearing my mind.
After a while, it seemed, I heard a tapping approaching. The steady tapping closed in, ending right in front of me. I opened my eyes. The figure before me was breathing slowly. I turned around and saw that the yellow now lighted the figure’s face.
“Are you alright?” a familiar voice said, emmitting from the figure.
I looked at the voice, but I didn’t respond. I felt paralyzed. The thoughts that had plagued me all night left my mind, leaving only the negative feelings. I wanted to communicate at least this, but the words escaped me. The numbness had overtaken me.
“Come here,” the voice whispered.
I noticed that the figure had outstretched its arms. Though my mouth was unable to form words, the rest of my body was responsive. I stood up steadily. I let out a sigh, closing my eyes and leaning towards the figured.
Instead, I felt a hard force pushing against me. I opened my eyes. The familiarity had suddenly gone away. The room became dark. The surface I lied on as solid. A frigidity started to set in. The darkness seemed to go on for miles. I felt incredibly in this bleakness. There was a swooshing sound, but no wind to accompany it. I knew this feeling. It was fear. I felt my heart beat faster, faster, louder, stronger, cruder, farther, longer, swifter, deeper, quicker, tougher, and stop.
I awoke to find myself breathing rapidly. I turned to look up, noticing I was home. I took a deep breath and tried to collect myself. I looked out the window, noticing that it displayed the grey of the sky.
I remembered what had been bothering me.