Decoding AI: Logic, Math, and Learning

Unraveling the Core Principles of Artificial Intelligence

Part 1: The Logical Debate on Intelligence: Philosophical and Historical Perspectives

The “underlying logic” of Artificial Intelligence (AI) is not a singular, fixed concept. Rather, it stems from a decades-long intellectual debate about how to create intelligence. To understand AI, one must first delve into its intellectual origins - the conflict and fusion of two core philosophical schools: Symbolicism and Connectionism. These schools represent distinctly opposing views of intelligence, and their fluctuating fortunes have shaped the historical trajectory and future direction of the entire AI field.

Two Schools of Thought

The construction logic of artificial intelligence unfolds along two main paths: top-down symbolic manipulation and bottom-up bio-inspired learning.

Symbolicism (The “Top-Down” Logic)

Symbolicism, also known as logicism or the computer school, is based on the core belief that the essence of intelligence lies in manipulating symbols according to a set of clear, formalized rules. This is a “top-down” approach, with the premise that human cognition and thought processes can be abstracted into symbolic operations. In this view, intelligence is seen as a process of logical reasoning, and the mind can be likened to a computer program running on structured data.

The most typical manifestation of this school is Expert Systems. These systems enjoyed their golden age in the 1970s and 1980s, marking the first large-scale commercial success of AI. They aimed to simulate the decision-making processes of human experts in specific narrow fields (such as medical diagnosis or chemical analysis) through a knowledge base containing a large number of “if-then” rules. The success of expert systems propelled symbolism to its peak, making it almost synonymous with AI at the time.

Connectionism (The “Bottom-Up” Logic)

In contrast to symbolism, connectionism, also known as the bionics school, argues that intelligence is an emergent phenomenon. It is not dominated by a central controller or preset rules, but rather arises from the complex interactions between a large number of simple, interconnected processing units (i.e., artificial neurons). This “bottom-up” logic is inspired by the structure of the human brain, believing that intelligence is not programmed, but rather obtained by learning patterns from data.

The core belief of connectionism is that complex behaviors can arise from simple local interactions, without the need for global explicit rules. Its core technological embodiment is Artificial Neural Networks (ANNs). These models learn complex relationships between inputs and outputs by training on large amounts of sample data and continuously adjusting the “weights” (i.e., connection strengths) between neurons.

The Pendulum of History: Rise, Winter, and Revival

The history of AI development is not one of linear progress, but rather resembles a pendulum swinging back and forth between symbolism and connectionism. This process profoundly reveals that the success or failure of a theoretical paradigm depends not only on the depth of its ideas, but also on the constraints of the technology and economic conditions of the time. The underlying logic of AI does not evolve in a vacuum, and its development trajectory is a direct result of the complex interplay between (1) mainstream philosophical thought, (2) available computing power, and (3) economic feasibility.

Early Advantages and the First AI Winter

In the early days of AI, connectionism showed greatpotential. However, in 1969, Marvin Minsky, a leading figure in symbolism, published the book Perceptrons, which became a key turning point in history. Minsky rigorously proved mathematically that the simple single-layer neural networks of the time (i.e., perceptrons) could not solve some of the most basic problems, such as the logical “exclusive or” (XOR) problem. This precise academic critique, combined with the general scarcity of computer computing power at the time, dealt a devastating blow to connectionist research. Research funding was drastically cut, and neural network research entered a period of stagnation lasting more than a decade, known as the first “AI winter.” During this period, the logic of symbolism occupied an absolute dominant position.

The Golden Age of Symbolicism and the Second AI Winter

Expert systems flourished in the 1980s, pushing symbolism to the peak of commercial applications. However, its limitations were gradually exposed: expert systems were expensive to build, knowledge bases were difficult to maintain, they could not handle ambiguous information, and they did not have the ability to automatically learn new knowledge. Ultimately, the commercial failure of “Lisp machines” specially used to run symbolic AI programs (such as the Lisp language) marked the end of this era. The rise of general-purpose computers (such as the IBM PC) with stronger performance and lower prices made these dedicated hardware devices uncompetitive, and the AI field then entered the second winter. This once again proves that if a theoretical logic is to continue to develop, it must have a strong and economical hardware foundation as support.

The Revival of Connectionism

The revival of connectionism was not accidental, but was driven by three key factors:

1. Algorithm Breakthroughs: During the “winter”, the introduction of backpropagation algorithms and the invention of more complex network structures such as long short-term memory networks (LSTMs) laid the algorithmic foundation for the effective training of neural networks.

2. Data Deluge: The popularity of the Internet brought an unprecedented amount of data. This data provided sufficient “nutrition” for neural networks that require a large number of samples for training.

3. Computing Power Revolution: Graphics processors (GPUs), initially designed for video games, have a massively parallel computing architecture that was found to be perfectly suited to the core matrix operations in neural networks. The emergence of GPUs broke the computing power bottleneck that had plagued connectionism for decades, allowing its theoretical potential to be truly unleashed.

Finally, the convergence of algorithms, data, and computing power ignited the deep learning revolution, making the logic of connectionism the undisputed mainstream in the AI field today.

The Philosophical Impasse: Understanding vs. Simulation

The historical dispute between the two major schools ultimately leads to a profound philosophical question that remains unresolved to this day: Does a machine capable of perfectly simulating intelligent behavior truly possess the ability to understand?

The Turing Test

Alan Turing’s “Turing Test” provides an operational, behaviorist definition of intelligence. The test involves whether a machine can have a conversation with a human, and the human cannot tell whether it is a machine or a person; then the machine can be considered intelligent. The Turing Test sidesteps the essential question of “what is intelligence” and turns to “what behavior should intelligence exhibit”.

The “Chinese Room” Thought Experiment

Philosopher John Searle proposed the famous “Chinese Room” thought experiment in 1980, launching a fierce attack on symbolism and the Turing test. The experiment is conceived as follows: A person who does not understand Chinese is locked in a room, and the room contains a detailed manual of Chinese processing rules (equivalent to a program). He receives notes with Chinese characters written on them (input) through a window, and then strictly follows the instructions in the rule manual to find and combine the corresponding characters, and then passes the results out of the window (output). To people outside the room, the room’s response is no different from that of a native Chinese speaker, so it passes the Turing test.

However, Searle pointed out that the person in the room never understood the meaning (semantics) of any Chinese characters from beginning to end, and all he did was pure symbolic manipulation (syntax). Searle concluded that simply manipulating symbols, no matter how complex, can never produce true “understanding.” This argument powerfully challenges the view of “strong AI” (i.e., the belief that a correctly programmed computer can possess a mind).

Today, modern AI represented by large language models (LLMs) can be seen as a super-upgraded version of the “Chinese Room” in a sense. They generate seemingly intelligent answers by statistically matching patterns in massive amounts of text data. The debate over whether they truly “understand” language or are just complex “stochastic parrots” is a continuation of the Turing vs. Searle debate in modern times.

For a long time, symbolism and connectionism have been regarded as two mutually exclusive paradigms. However, the “war” of history is coming to an end in the form of a synthesis. The underlying logic of the future is not an either-or choice, but a fusion of the two. This trend is reflected in the rise of Neuro-Symbolic AI. This field aims to combine the powerful pattern recognition capabilities of neural networks with the rigorous logical reasoning capabilities of symbolic systems, with the goal of building more powerful systems that can both learn and reason. For example, modern AI agents can call external symbolic tools (such as calculators, database queries) to enhance their own capabilities, which is a practical combination of neural models and symbolic tools.

In addition, the “Mixture of Experts (MoE)“ architecture in modern large language models also echoes the expert systems of symbolism in concept. The MoE model consists of multiple specialized “expert” sub-networks and a “gating” network, which is responsible for selecting the most suitable expert to handle each input. This is functionally similar to a symbolic system calling specific functional modules according to rules, but its implementation is entirely connectionist - through end-to-end learning and differential optimization. This shows that the underlying logic of AI is moving from opposition to complementarity, creating unprecedented powerful capabilities through fusion.

Table 1: Comparison of Basic AI Paradigms: Symbolicism vs. Connectionism

Part 2: The Universal Language of Modern AI: Core Mathematical Principles

Unveiling the mystery of modern AI requires realizing that its “underlying logic” is not human common sense or reasoning, but a precise and universal mathematical language. In particular, connectionism-dominated AI is essentially applied mathematics driven by “data, algorithms, and computing power.” The processes of intelligence generation, learning, and optimization can be broken down into the synergy of three mathematical pillars: probability statistics, linear algebra, and calculus.

The Mathematical Nature of AI

The core task of current artificial intelligence can usually be described as: finding an approximately optimal solution in a high-dimensional, complex problem space. Instead of solving problems by exhaustively trying all possibilities, it applies mathematical methods to find a good enough solution. Mathematics provides AI with formal modeling tools and scientific description languages, and is the cornerstone for building, understanding, and improving AI systems.

Pillar 1: Probability and Statistics - The Logic of Uncertainty

Probability theory and statistics provide AI with a theoretical framework for reasoning in uncertain environments and extracting patterns from data. AI models are essentially probabilistic systems that learn the underlying distribution of data to make predictions and decisions.

However, the emergence of big data poses a severe challenge to the foundations of traditional statistics. Traditional statistical theories, such as the law of large numbers and the central limit theorem, are mostly based on the assumptions that samples are “independent and identically distributed” (i.i.d.) and that the sample size n is much larger than the number of features p (i.e., p ≪ n). But in the era of big data, these assumptions are often broken. For example, in image recognition tasks, a high-resolution image may contain millions of pixels (features p), while the training dataset may only have tens of thousands of images (samples n), which leads to the “curse of dimensionality” problem where p ≫ n. In this case, it is easy to generate “pseudo-correlations” that invalidate traditional statistical methods.

The rise of deep learning is, to some extent, a response to this challenge. It provides a method for automatically learning effective feature representations from high-dimensional data without relying on traditional statistical assumptions. Nevertheless, establishing a solid statistical foundation for this new data paradigm is still a major mathematical problem that urgently needs to be solved in current AI research.

Pillar 2: Linear Algebra - The Logic of Representation

Linear algebra is the “universal language” of the AI world, providing basic tools for representing data and models. In neural networks, whether it is the input (such as the pixels of an image, the word vectors of text), the parameters of the model (weights), or the final output, they are all expressed as a numerical structure: vectors, matrices, or higher-dimensional tensors.

The core operation in neural networks, such as a neuron weighting and summing all its inputs, is essentially the multiplication of matrices and vectors. The reason why GPUs can greatly accelerate AI training is precisely because their hardware architecture is highly optimized to efficiently execute these large-scale parallel linear algebra operations.

Pillar 3: Calculus and Optimization - The Logic of Learning

The learning process of AI is essentially a mathematical Optimization problem. The goal is to find a set of model parameters (e.g., weights and biases in a neural network) that minimize the difference between the model’s predictions and the true answers. This difference is quantified by a Loss Function.

Gradient Descent: The Engine of Learning

Gradient Descent is the core algorithm for achieving this goal and is the engine that drives the learning of almost all modern AI models.

• Core Idea: Gradient descent is an iterative optimization algorithm that aims to find the minimum point of a loss function. This process can be figuratively compared to a person descending a mountain in dense fog. He cannot see where the lowest point of the valley is, but he can sense the slope of the ground under his feet. The most rational strategy is to take a small step along the steepest downhill direction at the current position, and then repeat this process.

• Specific Process:

  1. Initialization: First, randomly set an initial set of model parameters (weights and biases).

  2. Calculate Loss: Use the current parameters to have the model make predictions on the training data, and calculate the total error (loss) between the predictions and the true labels.

  3. Calculate Gradient: Use Partial Derivatives in calculus to calculate the Gradient of the loss function with respect to each parameter. The gradient is a vector that points in the direction of the fastest increase in the loss function value.

  4. Update Parameters: Move each parameter a small step in the opposite direction of its gradient. The size of this step is controlled by a hyperparameter called the Learning Rate (usually denoted as η). The update formula is: parameter_new = parameter_old − η × gradient.

  5. Repeat: Continuously repeat steps 2 to 4 thousands of times. Each iteration fine-tunes the model parameters, causing the loss value to gradually decrease. When the loss value no longer decreases significantly, the algorithm “converges” to a local or global minimum point, and the learning process ends.

• Algorithm Variants: Depending on the amount of data used in each iteration, there are many variants of gradient descent, such as Batch GD, Stochastic GD (SGD), and Mini-batch GD, which provide different trade-offs between computational efficiency and convergence stability.

Mathematics is the unifying language that connects all modern AI paradigms. Whether it is simple linear regression, complex support vector machines, or huge deep neural networks, the underlying logic of their learning is common: define a model, define a loss function, and then use an optimization algorithm (such as gradient descent) to find the parameters that minimize the loss function. This mathematical framework based on “loss minimization” is the true core logic of how machines learn from data.

The mathematical logic of AI also marks a fundamental shift from the traditional logic of programming. Traditional programming is deterministic and precise. AI, on the other hand, is probabilistic and approximate. As research has shown, the goal of AI is usually not to find a provably perfect solution (which is often impossible for complex real-world problems), but to find an approximate solution that is “good enough”. The “black box” characteristic of AI is a direct consequence of this shift. We can measure whether it is effective by evaluating its loss or accuracy, but it is difficult to explain how it works with step-by-step clear logic, as we can with traditional algorithms. This is because the “solution” of AI is not a set of human-readable rules, but a high-dimensional complex function encoded by millions of optimized numerical parameters. Its inherent “logic” is embodied in the geometric morphology of the multidimensional space formed by the loss function, rather than the semantic rules themselves.

Part 3: LearningMethodologies - How AI Acquires Knowledge

Building upon the core mathematical principles, AI has developed three primary learning strategies, or “learning paradigms.” These paradigms are categorized based on the types of data and feedback signals available to the AI system during training, namely: supervised learning, unsupervised learning, and reinforcement learning.

Supervised Learning: Learning with a Mentor

Supervised Learning is the most widely used machine learning paradigm.

• Core Logic: The model learns from a labeled dataset. In this dataset, each input sample is explicitly paired with the correct output answer. This process is like a student preparing for an exam with a set of exercises with standard answers.

• Learning Process: The model makes a prediction for an input sample, and then compares the prediction with the true label, calculating the error (loss). Then, optimization algorithms such as gradient descent are used to adjust the internal parameters of the model to reduce this error.

• Major Tasks and Algorithms:

  • Classification: Predict a discrete category label. For example, judging whether an email is “spam” or “not spam”, or identifying whether an animal in a picture is a “cat” or a “dog.” Common algorithms include Logistic Regression, Decision Trees, and Support Vector Machines (SVM).

  • Regression: Predict a continuous numerical value. For example, predicting the price of a house, or the temperature tomorrow. Common algorithms include Linear Regression and Random Forests.

• Data Requirements: The success of supervised learning heavily relies on a large amount of high-quality, manually labeled data. Obtaining this labeled data is usually costly and time-consuming, which is a major bottleneck for this method.

Unsupervised Learning: Learning Without a Mentor

Unsupervised Learning explores the intrinsic structure of data.

• Core Logic: The model receives unlabeled data and must autonomously discover hidden patterns, structures, or relationships in the data. This process is like an anthropologist observing an unknown tribe, without any guides, and can only identify different social groups and behavioral customs through observation.

• Major Tasks and Algorithms:

  • Clustering: Group similar data points together. For example, dividing customers into different groups based on their purchasing behavior. Common algorithms include K-Means and Gaussian Mixture Models (GMM).

  • Association Rule Learning: Discover interesting relationships between data items. For example, discovering the rule “customers who buy bread are also likely to buy milk” in market basket analysis.

  • Dimensionality Reduction: Simplify data by finding the most important basic features in the data while preserving most of the information. For example, Principal Component Analysis (PCA).

• Important Significance: Unsupervised learning is crucial for exploratory data analysis and is the cornerstone of the “pre-training” stage of modern large language models (LLMs), enabling them to learn general knowledge of language from massive amounts of unlabeled text.

Reinforcement Learning: Learning Through Trial and Error

Reinforcement Learning is inspired by behavioral psychology and is a paradigm for learning by interacting with an environment.

• Core Logic: An Agent takes an Action in an Environment and receives corresponding Reward or Punishment as feedback. The goal of the agent is to learn an optimal Policy, that