Life Lessons from a Guy who Invented Abstract Math
Have you ever been sitting in math class with partial derivative doohickeys flying all around you wondering how ANYONE ever even figured all that stuff out???
Your professor tells you another wacky principle every day… and most people just memorise it without thinking about it. But shouldn’t we ask whether they actually make sense? Often, we just accept things as true because someone tells us they are. I guess it’s easier, so we have to be glad SOMEONE bothered to figure it out all those centuries ago.
And in the case of your abstract math class, that SOMEONE was René Descartes. You know… *that* René Descartes. The “I-think-therefore-I-am” René Descartes. Apparently, he’s also the “let-me-just-casually-lay-the-foundations-of-all-of-graphing” René Descartes.
And because mathematicians LOVE puns so much, they decided to name his discovery the CARTESian plane 😆!!!
Anyways, the most important thing about Descartes was his obsession with figuring things out. Luckily for us, he left behind some tidbits from his experiences, so we can figure out how to figure things out for ourselves!
The Life of Someone Too Hard for Math Teachers to Understand…
Descartes was the closest person to a genius you can imagine. Better than asking what he did to deserve that is to ask what he DIDN’T do. He was a philosopher, metaphysician, physicist, mathematician, author, theologian, military strategist, AND royal life coach. *Y’know… like normal people.
The most important part of his work wasn’t the mathematics though *gasp* … it was the philosophy :-) Good old René marks the beginning of the ‘modern’ philosophers (early 1600s modern, that is 😕) He’s one of the major thinkers to challenge Aristotle (who apparently hadn’t been challenged enough over the course of several hundred years).
Until Descartes came along, a lot of people stuck to Aristotle’s ideas on empiricism (ie. SCARY word that just means using your senses). But Descartes instead preferred his logical conclusions and proofs, so he believed instead in reaching conclusions based on thinking instead of feeling.
So he started redefining the foundations of philosophy at the time with his new ways. Challenging the foundation of society’s knowledge is hard work though, so he had a pretty complex life. Moved around a bunch (once a year for 10+ years), accidentally irritated some holy Roman Popes, y’know… the usual. 😎
Because of all that outer complexity though, he had a reason to simplify his inner thoughts. So he left behind six meditations on some of his new theories!!! 🎉
*That might not sound exciting, but he wrote some of them in FRENCH
**That might not sound exciting either, but scholars wrote important ideas in Latin (which most people couldn’t read). So this was very NEW for commoners.
So what ARE these amazing theories that defied even Aristotle???
1. The Obvious Stuff: He thinks therefore he a̶m̶ is
Although this is part of every article about Descartes, it’s important to give context for all his other theories. The phrase gets inserted in some pretty random places in society, but there’s a more intellectual start to the story.
Descartes started a thought experiment which he recorded in his first few meditations. In it, he described himself waking up and wondering if he was really sitting in bed or if it was just a dream. He couldn’t tell because he’d had dreams in the past that felt exactly like real life.
So then he got thinking… how did he know if he had really woken up on any of those other mornings? Could those have been dreams? Descartes thought yes. ANY of those experiences he had felt through his senses could just be a realistic dream!
So Descartes being Descartes, he wanted to get to the bottom of the issue (which was whether he could trust ANY of reality because his senses weren’t always reliable 😨). He started doubting one experience after another… and just kept on finding more examples of his senses being wrong!
For instance, consider you’re Winnie the Pooh and you’ve just run across a bees’ nest!!! You get your (literally) sweet reward in the form of honeycombs, but then you stop. Today, you’re feeling a bit classy, so you don’t want honeycombs you want PURE. LIQUID. HONEY!!!!
So you leave your honeycombs by the fireplace, all brittle and cold and unclassy. A few minutes later, you come back and you finally see the warm, gooey treat you were looking for! 😋 BUT WAIT.
How do you know that dripping, molten liquid is the same as the solid honeycomb you left there??? It doesn’t feel, look, taste, or smell the same. If you just trusted your senses, the honey would seem like two separate things. But we can use our reasoning to identify it as the same old honey.
For Descartes, an example like that would show that we can’t always trust our senses, but our logical reasoning is more reliable. His conclusion was that he could doubt whether he had woken up, whether he was in his bed and so on… BUT there was some basic mental logic that he couldn’t doubt.
And that thing he couldn’t doubt… well it was that he was DOUBTING things in the first place!!! 🤯 It may not sound that important, but that changed Descartes’ outlook on life. Before, he thought that anything he ever felt through senses (ie. anything that exists) might not truly exist. Now, he realised that (at the very least) HE exists as a thing that can think and doubt.
That’s why he said the famous, “I think therefore I am.” It literally means that he couldn’t doubt that he existed because he was thinking. And this is the basis of all his lessons for us. He used this principle as ground zero for all his other theories — literally working from the MOST basic building blocks.
And those theories were standard stuff… y’know contemplating the existence of god, questioning the principles of geometry, creating physical laws of motion, etc. 😄
The only problem with his assumptions is when one of them failed the rest also went down the drain. Like you trying to figure things out in that math class 😉
So I won’t get into all his arguments, their counter-arguments, and the counter-counter-arguments to those counter-arguments 😌. But now you can understand where he’s coming from for the other abstract ideas.
2. So is there REALLY a big brother up there?
Descartes believed so. Of course back in the context he lived, he would kind of be killed if he didn’t accept Christianity soooooo… ✝✝✝✝ it is! Still, he had an interesting argument to justify that belief and its underlying idea applies to things beyond religion.
It all stemmed from his theory on where real things come from. Essentially, Descartes thought things that exist need to come from other things that exist.
*all the physicists in the room just walked out in disagreement* 😁
Putting the modern physicists aside for a second, that seemed to make sense.
A thought started from a brain, a table started out as a tree that was cut down, and a baby whale started out from two other whales conducting biochemical interactions 😉. Existing things came from other existing things. And in addition to that, the previous things are AT LEAST as real as the new things.
Take the first example of the thoughts and the brain, for instance. You could argue that mental thoughts aren’t as real as physical brains. In that case, the thoughts came from something MORE real than themselves (which is still AT LEAST as real as the thoughts).
*If the abstractness is too weird, just think about the whales again for a bit ;-)🐳
Then, Descartes tried to do something crazy. He thought about how that theory applied to himself and god. From earlier, Descartes concluded that he was a real thinking thing. And he came from his parents (also real thinking things). And they came from their parents and so on and so on…
But somewhere along the way, shouldn’t there be something MORE real and more complex? Like with the more concrete brain eventually creating abstract thoughts?
Ie. COMPLEX?????? → ……. → grandparents → parents → you
So what WAS that complex thing that everyone came from and that must be at least (if not more) real than us humans? Descartes’ answer was god. He believed that god existed as an inherent conclusion from this reasoning.
Of course, now that we have evolutionary biology that doesn’t really make sense. So if that argument doesn’t work for you (it didn’t for most philosophers), he had another one ready. And this one involved triangles (which of course, you’ll be familiar with from boring math class 😉).
See Descartes thought that a triangle wasn’t really a physical thing. Sure, you could have things shaped like triangles, but a triangle is really just a concept of a shape with three sides. It doesn’t have to be physical. And because triangles were concepts, all the properties triangles had inherently existed no matter whether we’re talking about a physical one or not.
For example, the angles in a triangle add up to 180 degrees. That’s a true property of ANY triangle, whether it’s something that physically exists or not. And if you said that property exists, then triangles ALSO have to exist.
I hope you’re not TOO lost, because it gets more complicated once Descartes involves god :-) Descartes tried to think about what people meant when they thought of ‘god.’ The one thing he found common among all ideas of ‘god’ was that ‘god’ = a perfect, all-powerful thing.
Ie. Being perfect and all-powerful is like an inherent trait of god. No matter how a culture/group defined god, the perfect all-powerfulness was like a triangle having 180 degrees. SO just like before, if the property of god being perfect and all-powerful truly exists, then god MUST truly exist.
…
*Philosophers throw even more of a tantrum with this one… 😂
Since there are lots of complicated holes to poke at here, I’ll only include one come-back (my favourite one) to Descartes:
“God, he says, either wishes to take away evils, and is unable; or He is able, and is unwilling; or He is neither willing nor able, or He is both willing and able.
If He is willing and is unable, He is feeble, which is not in accordance with the character of God; if He is able and unwilling, He is envious, which is equally at variance with God; if He is neither willing nor able, He is both envious and feeble, and therefore not God; if He is both willing and able, which alone is suitable to God, from what source then are evils? Or why does He not remove them?” — Epicurus
Anyways, the big point here is that Descartes took a simple theory he came up with and used it to think about some of the largest questions we have. AND he did it in a time where he could be killed for doing so 😮
3. So Now What I do with All these Gods, Whales, and Triangles???
Great question! It’s really bizarre trying to understand Descartes’ abstract ideas and how they apply to real life. Well, the biggest lesson is realising for yourself how to actually think.
Yes, you could just go back to tuning out from your math teacher during the next lesson and just memorise all those principles as true. OR you could actually start thinking about WHY all the knowledge we run into every day is actually true!
Don’t accept EVERYTHING as true without thinking about it (which doesn’t mean you can’t accept ANYTHING as true without thinking about it — otherwise you’d be stuck thinking all day 🤯). But I think most people definitely have room to grow in doing more of this.
Still, Descartes recognised that his kinda kooky theories were pretty out there. He told people at the end of his meditations not to use these theories as principles on how to live life… he had a separate list for those:
“To obey the laws and customs of my country, holding constantly to the Catholic religion, and governing myself in all other matters according to the most moderate opinions accepted in practice by the most sensible people.”
Translation: don’t live based on crazy theories. It pays to abide by the norms around you.
“To be as firm and decisive in action as possible and to follow even the most doubtful opinions once they have been adopted.”
Translation: don’t overthink things too much when it’s time to make decisions and be persistent in what you decide to pursue.
“Try to master myself rather than fortune, and change my desires rather than the order of the world.”
Translation: you’re in control of yourself, but not your luck. Make your inner desires match the world instead of trying to make the world match your inner desires.
“Review the various professions and choose the best.”
Translation: this is just to make you Millenials feel bad when reading this hundreds of years later 😅.
Overall, Descartes’ abstract theories on doubting everything and the conclusions he built up from those theories definitely seem weird. But the biggest lesson from this is his approach to finding knowledge by questioning things and making discoveries yourself.
In practical life, we’re bombarded with information left, right, and centre, but we don’t stop to think about it. So instead, what if the next time your math teacher is lost in complex graphs, you told them, “How can YOU know that works when the guy who invented it didn’t???”
Interesting tidbits:
- He *intensely* debated with Pascal about whether it was possible to have vacuums (the physics kind, not the cleaning kind) 😕
- He slept from 12 AM to 12 PM. I don’t know if you call that late or early, but he IS a genius so maybe he could do both 😁
- His methods for approaching geometry, graphing, and equations are still hated by students to this day… 😱
Key Takeaways
- Descartes’ work extended to more professions than I can list and it still influences academia and culture today.
- His logical theories on how all senses can be doubted and whether something really exists changed Western philosophy during his time.
- The biggest actionable from learning about Descartes is to take his ideas to heart and actually start thinking for yourself.
Before You Go
Hey, I’m currently trying to think for myself in figuring out how to get done the giant stack of work for this weekend! If you like this article, feel free to:
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To join forces in thinking for myself *collaboratively* 😅 ;-)
