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Some things are just obvious. The sky is blue, the letter ‘A’ comes before ‘B’, and 1+1 equals 2 except that the latter isn’t as obvious as you might think. Think about it. What does it mean when we say that we’re adding stuff? What does it mean when we say two things are ‘equal’? What does it mean when we talk about the number 1? Here’s a fun little exercise for you: Try to describe what a number is without using the word ‘number’ or any other word/concept derived from the word ‘number’.
Now, if you really put all of your brain cells at work trying to accomplish the above task, it’s time to giddy your horses up. This is gonna be a wild ride as we venture deep into the world of pure logic and find an abyss that hides beneath the foundations of reality!
The Foundation of Mathematics: Logic
We learn about science by observing the world around us. We observe, theorize, and experiment to uncover natural phenomena. It seems pretty fascinating to imagine a universe where time moves backward, or where string theorists acknowledge experimentalists. However, when you try to think of a universe where 1+1 equals 3, it doesn’t feel right. It seems True by definition and even though math works exceptionally well when it comes to describing our universe, is it really about our universe? This begs the question, Of course, we aren’t the only souls to have thought of it. This very question captured the attention of one of the greatest philosophers of all time, Plato. He thought of numbers, geometrical shapes, and the relations between them as objective truths. Regardless of what world they’re in, they are bound to be true. In Plato’s view, a ‘World of Forms’ existed independently where objects like numbers were present. However, one of his pupils, Aristotle, had contrasting views. He rather believed that numbers were properties of objects than numbers themselves being objects. For instance, if a human is said to have 2 eyes, it doesn’t mean that ‘2' and ‘eyes’ are separate objects. He thought of the number 2 as a property used to describe a pair of eyes. Numbers describe features of our world. Philosopher Immanuel Kant thought of them to be described by intuition and experience. WAIT, HOLD UP! You just used ‘2’ to describe ‘1’ pair of eyes!?
If numbers correspond to an object’s property, how can multiple numbers be used to describe the same object? It doesn’t make sense. This thought daunted German mathematician, logician, and philosopher Frege (Friedrich Ludwig Gottlob Frege). He reasoned that numbers don’t actually apply to objects but to concepts. He strongly believed that mathematics (especially arithmetic) can be reduced to pure logic i.e., you can understand math based on reason alone. Logic is a tool for reasoning about how different statements affect each other solely through deduction and inference. This idea is famously called Logicism.
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Why 1+1=3 | क्यों 1+1=2 को prove करने में 379 pages लगे | 379 page proof that 1+1=2
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