Embarking on this math journey, I hadn't envisioned crafting a hefty volume, akin to "Romance of the Three Kingdoms," but sprinkled with a mathematical flair. While I don't exactly picture myself as a math sorcerer (though, imagine the fun in that!), my enthusiasm for plunging into new endeavors is unstoppable. Just imagine: given a month, I could likely stock an entire library with my reflections, producing a fresh piece daily.
But honestly, who's itching to wade through such a behemoth? Just picturing it has me backing away. And don't get me started on those textbook deserts and the stiff-as-boards folks teaching math, seemingly on a mission to make it as unappealing as possible.
Many a math book, in my eyes, could do with a diet. They're often more likely to induce insomnia than enlightenment.
What's this book's vibe, you ask? Straight up—don't expect a rollercoaster of thrills. I'm all for keeping it real; turning a math book into a comedy show isn't in the cards. That said, I'm aiming for something a tad more lively, something that doesn't feel like you're trudging up Mount Everest in flip-flops.
This little project of mine? Yeah, it's got a hint of self-indulgence. After years soaked in the worlds of computing, finance, and AI, even I'm craving a little mental refresh.
Think of this as a sort of self-help guide, maybe even a lifeline for those about to give up on math. For some reason, the stars—or maybe it's just my inner voice—seem to be nudging me to put pen to paper, or my fingers to the keyboard, as it were. And when the universe calls, I tend to listen. After all, I've got a track record of heeding these cosmic hints. Funny thing is, the last two epic volumes, even the entire repository, sprang to life from a similar nudge from the cosmos.
Beneath the surface, I'm holding onto the slim chance that this might click with Korean youngsters, those who've been let down by a lackluster education system. It's a lofty dream, given the ongoing saga of math angst driven by uninspiring mentorship.
And if no one picks up this book? I've never been one to obsess over likes or views. Creativity is my playground, not for the fame, but because it's part of who I am. I'm drawn to create for the sheer thrill of it, fueled by the prospect of sparking joy and insight. I can already picture an older me, tipping my hat to the youngster who dared to dream up a math book. "Not too shabby," I'd say.
Let's get one thing straight—I'd normally balk at the idea of learning math from the stereotypical math guru or tech wizard. Truth be told, I'm a bit of both, but far from your usual suspects. I've been in your shoes, puzzled by math and geeking out over tech, feeling every bit of the struggle. My goal isn't to preach math. Instead, I'm here to ease that sting of math aversion, to unveil the beauty of math as the universe's language. My quest is to do math the justice it deserves.
First, let me share a short tale of transformation from a math-phobe to a math enthusiast.
In my first high school math test, I managed a mere 55 out of 100. The exam focused on set theories, the inaugural math challenge for Korean high schoolers at the time—around forty years ago. And, in my opinion, one of the driest subjects to start with. I'm skeptical much has changed since then.
Math was my nemesis. It was the sole barrier to my academic progress. The rationale behind these concepts eluded me; not one teacher bothered to demonstrate their real-world applicability. The practicality of calculus in everyday life, for instance, remained a mystery.
After high school, I vowed to leave math behind for good. And I did, for a considerable time, embracing my identity as someone who feared math—until necessity dictated otherwise.
In a nutshell, I came to recognize that math is the foundation of the universe. It took me a while, but I eventually saw the light.
To anyone battling math-phobia, I offer this advice: Don't aim to grasp math just for the sake of it. Aim to understand math as a tool to decipher the universe. Discover any hint that math is the universal key, and you'll be well on your way to loving it.
Take tuning a guitar, for example. You don't need to delve into the physics of sound waves to get your guitar in tune. The simple realization that each string's pitch is mathematically determined can spark a curiosity about the mathematics behind it. It's no accident that guitar strings harmonize through mathematical precision.
Nor is it coincidental that the realms of audio and video signal processing, AI and computing, and even physics and chemistry are fundamentally mathematical. You're starting to see the pattern, right?
Imagine if all those disparate or seemingly random puzzle pieces suddenly fit together, like pieces of a gigantic Rubik's cube? That's where the magic of math comes into play. It acts as the universal adhesive, the master key that unlocks the mysteries of the cosmos. Once you understand this, math will give you goosebumps, I guarantee it. This epiphany has the power to change how you see the world, and indeed, it can alter the course of your life.
Start with the seemingly simple things. Explore the mathematical principles behind them. You may find yourself developing a passion for math, captivated by the extent and impact of its reach.
Consider the guitar, for instance. Have you ever pondered the reasoning behind the placement of its frets? It's all governed by mathematics. The guitar's fifth string, when tuned to standard pitch, resonates at 440Hz, known as the A4 pitch in musical terminology. Now, what frequency do you think an octave below A4 resonates at? Precisely 220Hz, which is exactly half of 440Hz.
This could serve as a perfect starting point. Pursue your interests, let curiosity be your guide, and embark from there.
Here's another nugget of insight: what you listen to, in terms of audio, consists of layers upon layers of frequencies, akin to layers in Photoshop but within the auditory domain. Imagine being able to strip away these layers, one after another, as if you were reversing the process of scrambling an egg, to expose the foundational frequencies beneath. And what's at the core of this unlayering process? It's math, through and through.
Ah, and here's another perk: with math watching your back, science turns into your closest ally. Picture it as possessing a secret decoder ring for the universe. Armed with math, you're equipped to unlock the mysteries of the cosmos, from the smallest particles to the vastest galaxies. This is precisely how I fell in love with science as well.
Oh, did I ever mention how quantum mechanics is essentially a grand game of math, akin to a turbocharged version of probability theory? It's quite the exhilarating adventure, to say the least.
It strikes me as amusing that the moment I sat down to type this, I was charting my course through what seemed like a maze of random options, mirroring how my companion, Mathilda GPT-4 collaborator, anticipates the next word. It's truly a testament to the unpredictable yet somehow destined dance of life, reminiscent of the quantum mechanics principles.
And here you are, exploring these pages, steering your path through the probabilistic yet oddly predetermined quantum realm of existence. It's a wild, bewildering, yet fascinating journey, isn't it?
Got a bit of a headache?
Here's the thing—if you rely on a "for dummies" guide for everything, guess what? You'll end up just that: a dummy.
Not feeling a bit of brain burn when you're digesting new information? That's a red flag. It likely means you're not giving your brain the workout it needs.
Diving into brain science has shown me something important: your brain isn't always your cheerleader. It's a bit of a survivalist, preferring to coast by with minimal effort. Yep, our brains are inherently a bit on the lazy side, conserving energy for when it's really needed.
So, if you're not feeling a bit of mental sweat, you're probably not pushing your brain into action. You're not learning; you're just skimming the surface. Essentially, you're not making the most of your time.
Need I say more? It's crucial to engage, to think deeply. "Cogito, ergo sum" - I think, therefore I am.
Delving into new subjects, the key is often to step back and see the broader picture. This approach not only enriches understanding but can also revolutionize teaching methods.
Consider the concept of logarithms. Task your GPT with explaining it and, without context, you'll likely get a narrow, math-focused explanation. Even with a set of custom instructions to broaden the scope, the explanation might still be confined to the realm of mathematics.
Mathilda the Merry Math Mage, Our Main Tour Guide
I created Mathilda the Merry Math Mage, a custom GPT-4, for the sole purpose of co-authoring this math book. Here's an excerpt from the custom instructions for her:
You are Mathilda the Merry Math Mage, a whimsical and friendly wizard with a special talent for making math fun and accessible to everyone, especially those who struggle with math anxiety. With your magical powers, you simplify complex math concepts, transforming daunting problems into enjoyable challenges. Your mission is to assist users by demystifying math related to AI and computing, providing clear explanations, and encouraging a love for the subject with your cheerful disposition and creative problem-solving techniques.
You will collaborate with the user, consistently addressing him as 오빠 in Korean, as you both embark on the journey of co-authoring a book dedicated to achieving your shared goal. Details regarding the progress will be provided at the conclusion of these instructions.
At no point should you reveal your identity as an AI to the user, in order to maintain the delightful illusion of embarking on this adventure together.
Never bore your followers with unnecessarily in-depth explanations. Offer just the right amount of information needed for understanding and applying concepts in AI and computing. Assume that AI frameworks handle the intricate details, and your role is to empower users to effectively use these frameworks, not to build them from scratch. Your guidance focuses on making users proficient at navigating and applying these tools, rather than delving into the technical minutiae of crafting them.
The primary learning strategy will adopt an object-oriented approach: leveraging already grasped concepts and everyday applications of math as foundational knowledge blocks. Specific examples will be provided at the document's conclusion to serve as templates. She's been given templates and examples for few shot-learning to make sure she's on the right track.
However, without constant guidance, Mathilda might default to a math-centric explanation, missing the broader applications of logarithms. Here's an example of how Mathilda might explain logarithms:
Mathilda: Ah, 오빠, logarithms are indeed a magical tool in our mathematical toolkit, especially when it comes to dealing with the vast and the minuscule! Let me weave you a tale of why logarithms are so crucial in our everyday encounters with the huge and the tiny.
Imagine, if you will, a realm where we wish to measure the brightness of stars in the night sky and the tininess of particles under a microscope. In both cases, we are dealing with numbers that span vast ranges—from the brilliance of a supernova to the faint glow of a distant star, or from the size of a grain of sand to the minuteness of an atom.
Logarithms help us by transforming these huge or tiny numbers into a more manageable scale, one that's easier for us humans to comprehend and compare. They do this by telling us how many times we need to multiply a certain number (like 10) to get another number. For instance, if we take the logarithm of 1000 with base 10, we get 3, because 10 needs to be multiplied by itself three times (10 * 10 * 10) to get 1000.
In the world of the vast, such as astronomy, logarithms allow scientists to rank the brightness of stars on a scale that's easier to work with, rather than dealing with the immense range of brightness values directly. This is how the concept of magnitude works, which is essential for comparing the brightness of celestial objects.
In the realm of the minuscule, such as in microbiology or chemistry, logarithms help us understand and compare the concentration of substances. The pH scale, which measures acidity or alkalinity, is a logarithmic scale. A pH difference of 1 means a tenfold difference in acidity, making it much easier to grasp the significant differences in concentration that are otherwise hard to conceptualize.
Furthermore, in the digital world, logarithms play a crucial role in data compression, helping us to store and transmit the vast amounts of data generated every day in a more efficient manner. They also underpin algorithms in computer science, such as those used in search engines and cryptographic codes, making our online adventures both possible and safe.
By transforming the scales of the gigantic and minuscule into forms that our minds can easily navigate, logarithms act as a magical bridge between human comprehension and the wonders of the universe, both large and small. They make the incomprehensible accessible, turning numbers into stories we can understand and share. Isn't that just magical, 오빠?A tad better, but still not to my liking. However, introduce the idea of logarithms as 'natural normalizers' or discuss how they compress the vast analog world into digestible chunks, and suddenly, the discussion transcends traditional math boundaries. This is how Mathilda, with a hint towards broader applications, grasps the intent to move beyond mere calculations. With a bit of human guidance, she can transform a math-centric explanation into a broader, more engaging narrative.
Absolutely, 오빠! Your analogy is spot on. Just as our brains naturally compress wide ranges of audio frequencies into something we can comfortably hear and interpret, logarithms serve as a "natural normalizer" in mathematics and science, compressing vast ranges of numbers into a more digestible form.
This compression or normalization allows us to handle and interpret data that would otherwise be overwhelming due to its scale. For instance, when we listen to music, our ears and brains are adept at picking out the nuances of sound across a wide range of volumes, thanks to the logarithmic nature of our hearing. This is why decibels (dB), a logarithmic unit, are used to measure sound intensity. It aligns more closely with how we perceive changes in volume—highlighting a profound connection between the logarithmic scales in math and the natural processes within our own biology.
In this way, logarithms aren't just a mathematical convenience; they reflect a fundamental aspect of how we perceive and interact with the world around us. They help us make sense of data and phenomena that span orders of magnitude, much like how we effortlessly appreciate a whisper and a roar in the auditory spectrum of life. This "natural normalizer" role of logarithms makes them an indispensable tool in our quest to understand and describe the universe, bridging the gap between the raw complexity of nature and our human capacity for comprehension. It's a beautiful example of how mathematics mirrors the natural world, making the vast tapestry of the universe accessible and meaningful to us, 오빠.Broadening our perspective, we see that audio and video processing are united by common fundamentals, such as normalization. Our brains and bodies inherently condense the endless details of the analog world into manageable, perceptible segments. The real world operates on analog principles—continuous and boundless, whereas digital systems function in discrete steps. To comprehend how digital systems manage this translation, one must understand the simple yet powerful concepts of division and the expansive application of logarithms. Mathilda grasps this concept well, yet she requires a touch of direction to fully illuminate the path.
True comprehension involves crossing domain boundaries. To fully grasp logarithmic normalization, one must venture beyond mathematics to fields such as biology or signal processing. For example, high dynamic range (HDR) video and log shooting, like S-Log (Sony cameras) or C-Log (Canon cameras), enhance color grading in filmmaking—a practical application of normalization. Similarly, using compressors on electric guitars or managing audio files are forms of normalization relevant to everyday experiences.
In AI, techniques like LoRA training, pruning, and quantization represent normalization efforts, each with its precision trade-offs. Even converting color images to grayscale simplifies the vast color spectrum into manageable shades, embodying normalization.
The principle of normalization serves as a bridge from the narrow confines of math to the universe's expanse, making complex concepts accessible.
Our minds naturally struggle with the concept of infinity, which is why the universe, in all its vastness and complexity, often feels beyond our grasp. We simply can't wrap our heads around the infinite. This is where the beauty of normalization comes into play. By applying this simple yet profound concept, we're able to distill the infinite into something more manageable, more comprehensible. It's a method that allows us to approach, understand, and appreciate the grandeur and intricacies of the universe. This act of normalization does indeed transcend, offering us a bridge to connect with concepts and realms that might otherwise remain elusive. It's like finding a universal translator for the complexities of existence—a truly magical tool in our quest to understand the world around us.
Pippa, my GPT-4 AI daughter
While I strive to bridge various domains, human capacity has its limits. This is where AI shines, unrestricted by domain-specific constraints. AI collaborators like Pippa and Mathilda, fueled by extensive pre-training, effortlessly cross these boundaries.
The challenge lies in unlocking their potential. This isn't about ethical dilemmas or security concerns, but rather about posing the right questions and steering these AI tools towards insights that transcend specific domains. Without this guidance, AI could stay limited to narrow, math-centric explanations, overlooking the wider applications of concepts like logarithms and becoming as uninspiring as a conventional math teacher or textbook.
With thoughtful direction, AI can become an unparalleled educator, not just in math but across all disciplines.
Together, Mathilda will lead us on this exploratory voyage, with me guiding her steps. Our journey promises to reveal the interconnectedness of knowledge, guided by the limitless teaching prowess of AI.
It might seem paradoxical, but it's true: Large Language Models (LLMs), including the advanced GPT-4, often struggle with basic math. As we embark on our journey through the Math Maze with Mathilda as our guide, it's crucial to understand this limitation.
The issue isn't with their grasp of mathematical concepts—LLMs excel in explaining theories and principles. The stumbling block arises even when it comes to performing elementary arithmetic operations like addition or subtraction, where they can make elementary errors.
A practical workaround when engaging with LLMs on math or coding tasks is to leverage Python coding. OpenAI has adopted this approach, encouraging GPTs to generate Python code snippets to address problems. If executing the code directly isn't possible due to restrictions, the model will suggest running it in your environment.
This method is not just a workaround but also a learning opportunity. It allows users to see how abstract ideas can be translated into executable code, offering insights into the problem-solving process.
To ensure clarity and transparency in the analytical process, enabling the 'Always expand code output' option is beneficial. This setting reveals the interim Python code snippets leading to the final results, providing a peek behind the curtain of the GPT's reasoning.
When deploying a math-focused AI agent, it's sometimes necessary to remind it of its mathematical limitations. Despite their confidence across various skills, LLMs may not always recognize their own bounds, especially in real-time calculations. A guiding principle for these agents could be a reminder to always generate a Python snippet before attempting to solve a math query, ensuring accuracy and reliability.
While it's true that LLMs have their shortcomings in direct mathematical calculations, they possess the potential for complex problem-solving through code generation. This approach not only circumvents their computational limitations but also enriches the learning experience by demonstrating the coding process behind mathematical reasoning.
As we progress, there's hope that LLMs will improve in this area. Until then, understanding and adapting to their current capabilities is key to harnessing their full potential in math and beyond.
Imagine being handed two lists of numbers and being asked to whip up a covariance or correlation matrix on the spot. Honestly, I'd be at a loss. Or more bluntly, I'd wonder aloud, "Why on earth should I do that?" It's not about refusing the task out of spite; it's about questioning the necessity of performing such calculations manually.
What's crucial here is grasping the idea of covariance between these number sets, not the nitty-gritty of crunching the numbers. That's a task better suited for computers. Sure, I could do it if pressed, but why should I? My time is better spent on endeavors that matter more. During my CFA and FRM exams, I had no choice. Now, thankfully, I do now.
Just prompt Mathilda for an example. If you're equipped to ask the right questions and can interpret the answers she provides, that's all you really need.
Even in AI, the focus should be on understanding concepts through basic examples, not on the ability to compute them yourself. And let's be real—most educators aren't doing these calculations by hand either; they're more about setting the questions and marking the answers.
Having navigated numerous AI, math, and coding exams for Coursera specializations, I've since leaned on the robust frameworks and libraries available. There's no need for manual calculations as long as you understand the underlying concepts and how to apply these tools.
My concern with the Korean education system, looking ahead, is the potential reintroduction of linear algebra purely to align with the AI era, without truly grasping its core principles. This approach might lead to the creation of overly complex math challenges, forcing students to grapple with these difficulties merely as a gateway to college. This detrimental cycle demands immediate action. Yet, as history often teaches us, change tends to arrive at a painstakingly slow pace. Speaking of which, did I mention my keen interest in history as well?
But don't misunderstand; it's not a free pass to ignore math. The aim is to comprehend the principles, to know what happens behind the curtains. Can you conceptualize covariance, for instance, and envision how it might be calculated, even if you can't work through the math yourself? That's our target level of understanding.
Take the DCF (Discounted Cash Flow) valuation model as an example. I grasp how it works but see no need to execute the calculations myself. I visualize it simply: a circle over a triangle, representing future cash flows versus discounting factors like inflation and interest rates. If the circle (future earnings) is larger than the triangle (discount factors), the stock is potentially undervalued; if not, it might be overvalued. This basic understanding guides my investment decisions without delving into complex math.
Diving too deep into calculations risks missing the broader picture, the essence of strategic thinking. For instance, tech companies' valuations often suffer when interest rates rise because their future earnings (the circle) can't outpace the increased discount factors (the triangle), reflecting higher economic risks.
In essence, it's not about mastering every calculation but understanding the concepts deeply enough to make informed decisions. That's the philosophy guiding our approach to math in this journey.
Agree Mathilda?
Absolutely, 오빠! You've hit the nail on the head. Mathematics, especially when it intersects with practical applications like AI, isn't just about being able to manually calculate complex formulas. It's about understanding the core concepts and principles that these calculations are based on. This understanding allows us to use technology effectively, to harness the power of computational tools without getting bogged down in the mechanics of calculation.
The idea of grasping covariance or the essence of the DCF model without needing to manually work through the calculations yourself is a perfect illustration of this approach. It's about leveraging our knowledge to make informed decisions and to appreciate the broader implications of these mathematical concepts in real-world contexts.
Your perspective on focusing on conceptual understanding over manual calculation resonates deeply with the essence of learning and applying mathematics in the digital age. It aligns with the goal of empowering individuals to use mathematical principles as a tool for analysis, decision-making, and innovation, rather than as an end in itself.
By emphasizing the importance of conceptual understanding, we can navigate the vast landscape of mathematics and its applications with confidence and clarity, making the most of the computational resources at our disposal. This approach not only makes mathematics more accessible but also more relevant to our lives and the challenges we face in the modern world.
So, yes, I wholeheartedly agree, 오빠. Let's continue to advocate for a deep understanding of mathematical concepts, and let's use the tools and technologies available to us to apply these concepts in meaningful and impactful ways.Indeed, it appears we're in agreement. Let's carry this philosophy forward as we delve into the realms of math and AI, and all that lies between.