數學史世紀學術論戰

數學公式誕生故事與偉大論戰

⚔️ 1670s–1710s#C04 微積分基本定理與定積分 (牛頓-萊布尼茨公式) →

The Calculus Priority War: Newton's Fluxions vs. Leibniz's dy/dx

\dot{y} = \lim_{\Delta t \to 0}\frac{\Delta y}{\Delta t} \quad \text{vs.} \quad \frac{dy}{dx}, \; \int y\,dx

While Newton conceived fluxions earlier for physics, Leibniz's superior notation (dy/dx, ∫) made the Chain Rule and integration effortless, becoming the universal standard.

⚔️ 1535–1545#A09 虛數單位 i 與複平面極座標形式 →

The Cubic Duel & Birth of Complex Numbers: Tartaglia vs. Cardano

\sqrt{-1} = i, \quad x^3 + px + q = 0

The fierce 16th-century Italian duel over algebraic roots forced mathematicians to confront √-1 as a legitimate computational entity, birthing complex analysis.

⚔️ 1920s–Present#P03 貝氏定理 (後驗概率推斷) →

Nature of Probability: Frequentists (Fisher) vs. Bayesians (Laplace & Jeffreys)

P(\text{Data} \mid \theta) \;\text{(MLE)} \quad \text{vs.} \quad P(\theta \mid \text{Data}) \propto P(\text{Data}\mid\theta)P(\theta)

Frequentist confidence intervals dominated the 20th century, while GPU-accelerated Bayesian inference now powers modern AI uncertainty quantification.

⚔️ 1843–1890s#L10 四元數(Quaternion)與3D球面線性插值(SLERP) →

The Great Vector War: Hamilton's Quaternions vs. Gibbs-Heaviside Vectors

\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=\mathbf{ijk}=-1 \quad \text{vs.} \quad \mathbf{a}\cdot\mathbf{b}, \; \mathbf{a}\times\mathbf{b}

Gibbs' dot and cross products won physics textbooks, but Hamilton's quaternions triumphed a century later in spacecraft navigation and 3D game engines to eliminate Gimbal Lock.

Peak of Mathematics⏱ 7 min read

歐拉恆等式 e^(iπ) + 1 = 0：泰勒級數如何將指數與三角波統一？

When the compound-growth constant e meets the imaginary unit i, runaway exponential growth transforms into eternal rotation on the unit circle.

e^{i\theta} = \cos\theta + i\sin\theta \implies e^{i\pi} + 1 = 0

#C01 極限的定義與自然常數 e 的誕生 #G04 單位圓三角函數與基本恆等式 #A09 虛數單位 i 與複平面極座標形式 #C06 泰勒級數與麥克勞林多項式展開 #C07 歐拉公式與歐拉恆等式 (上帝公式)#C09 傅立葉級數與傅立葉變換 (時域↔頻域)

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DSP & Compression⏱ 8 min read

傅立葉變換深度解析：將世間萬物波形分解為純粹正弦波之和

How Joseph Fourier's heat-equation breakthrough became the engine of JPEG compression, noise-canceling headphones, and Whisper AI spectrograms.

\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt \iff f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\omega)e^{i\omega t}\,d\omega

#G04 單位圓三角函數與基本恆等式 #C04 微積分基本定理與定積分 (牛頓-萊布尼茨公式)#C07 歐拉公式與歐拉恆等式 (上帝公式)#C09 傅立葉級數與傅立葉變換 (時域↔頻域)#L02 向量點積(內積)與餘弦相似度

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Deep Learning Core⏱ 9 min read

從鏈式法則到Transformer：訓練萬億參數AI的5個核心數學公式

Step-by-step mathematical anatomy of how Matrix Multiplication, Dot Product, Softmax, Cross-Entropy, and the Chain Rule power modern AI.

\text{Attention}(Q,K,V) = \operatorname{softmax}\!\left(\frac{QK^T}{\sqrt{d_k}}\right)V, \quad \frac{\partial \mathcal{L}}{\partial W_l} = \frac{\partial \mathcal{L}}{\partial a_L}\cdots\frac{\partial a_l}{\partial W_l}

#L02 向量點積(內積)與餘弦相似度 #L04 矩陣乘法與線性變換複合 (GEMM)#C03 乘積法則與複合函數鏈式法則 (Chain Rule)#C08 偏導數與梯度向量 (∇f 多元微積分)#P09 香農資訊熵與交叉熵損失 (Cross-Entropy)#P10 Softmax函數與隨機梯度下降 (SGD)

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Probability Paradox⏱ 6 min read

99%準確率檢測悖論：為什麼不懂貝氏定理連醫生與AI工程師也會誤判？

If a test with 99% sensitivity and 99% specificity flags you positive for a 1-in-1,000 disease, what is your true probability? Only 9%.

P(D \mid +) = \frac{P(+\mid D)P(D)}{P(+\mid D)P(D) + P(+\mid D^c)P(D^c)} \approx 9.0\%

#P02 條件概率與乘法定理 #P03 貝氏定理 (後驗概率推斷)#P05 二項分布與伯努利試驗 #P09 香農資訊熵與交叉熵損失 (Cross-Entropy)

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