Mathematics: The Universal Language

Akshay Mestry

DePaul University Chicago, IL

xa@mes3.dev

0 mins read

In the realm of Artificial Intelligence, be it in its entirety or dissected into components such as Machine Learning, there exists a profound reliance on the tools borrowed from the world of mathematics. My journey into understanding Machine Learning and the broader realms of AI taught me that the mastery of mathematical principles is not just beneficial, but essential. Thus, as I pen this chapter, I aim to demystify the common mathematical notations and emphasize their pivotal roles in the AI domain. This segment, at its heart, embodies the notation aspect of L.E.A.R.N, designed to lay a foundational bedrock for those who, like me at the outset, may find themselves at the threshold of this vast knowledge domain without a map.

1   Introduction

I feel mathematics emerges not merely as a tool of inquiry or some kind of science but as the very dialect through which the universe reveals its secrets. While pursuing my graduate degree, one of my favorite college professors, Dr. Schaefer once said…

Mathematics is a universal language.

The above assertion by Dr. Schaefer, speaks truth far beyond the mere symbolic representation of numbers and equations. It is an acknowledgment of the fundamental role mathematics plays in bridging various disciplines, cultures, and even realms of thought. Apologies for being so philosophical, but from the intricate dance of celestial bodies, charted by astronomers, to the quarks and leptons that play hide and seek within the infamous quantum realm, mathematics offers a dictionary of precision and clarity. It is a language devoid of ambiguity, where each theorem, each equation, carries a meaning that is not altered by the nuances of human language. In this way, mathematics becomes the common ground on which scientists, engineers, and philosophers stand, enabling a dialogue that transcends linguistic and geographical boundaries.

  Yet, to say that mathematics is a universal language does not merely point to its role as a facilitator of communication across disciplines. It is also to recognize the innate human capacity to understand and manipulate abstract concepts, to see patterns in chaos, and to derive meaning from the seemingly impossible things. This language, with its numbers and symbols, speaks to a shared human experience, an underlying harmony that courses through the fabric of reality. As I go deeper into the realms of Artificial Intelligence and Machine Learning, I find this language ever more crucial. AI, in its essence, is an embodiment of human ingenuity, a mirror reflecting our desire to decode the complexity of our world. Through these mathematical models and algorithms, we teach machines to learn, to perceive, and to make decisions. Mathematics, in this context, is not just a tool but a bridge between human intelligence and artificial cognition, a means through which we inspire machines with the ability to understand and interact with the world.

2   Lexicon of Artificial Intelligence

2.1   Crafting the Vernacular of the Cosmos

In the rich tapestry of human knowledge and endeavor, mathematics stands as a beacon of clarity and precision. It is through this lens that I wish to explore the profound assertion…

We use signs and symbols to speak the language of maths and Artificial Intelligence knows only this language.

This statement not only highlights the intrinsic role of mathematics in the realm of Artificial Intelligence but also underscores the universal medium through which we communicate complex ideas and innovations. Mathematics, with its elegant array of signs and symbols, serves as humanity’s most refined tool for articulating the nuances of the universe. These symbols, steeped in centuries of human thought and discovery, are the alphabets of a language that knows no borders, a language that whispers the secrets of infinity and the microscopic intricacies of the quantum world alike. Each symbol, each sign, carries within it a world of meaning, a distilled essence of human understanding that transcends cultural and linguistic divides. In the sphere of Artificial Intelligence, this language finds its most ardent disciple. AI, in its myriad forms — from neural networks that mimic the neural pathways of the human brain to algorithms that sift through data with unerring accuracy — relies wholly on the language of mathematics. It is through this language that we instruct machines, encode algorithms, and define the very parameters of AI’s learning and functioning. The precision of mathematics allows for the creation of AI systems that can learn from data, identify patterns, and make decisions with a level of accuracy that often surpasses human capability.

2.2   From Symbols to Sentience

The journey of AI, from its early stages to its current state of flourishing quasi-sentience, is a testament to the power of mathematical language. We use mathematics to breathe life into machines, to endow them with the ability to learn, to reason, and to interact with their environment in ways that were once the sole purview of living beings. This process, intricate and complex, is facilitated entirely through the use of signs and symbols that comprise the language of mathematics. It is as if we are imparting a piece of the universal consciousness to these machines, teaching them to understand the world through the same numerical and geometrical constructs that have enabled humanity to reach the stars.

Tweet from @danishtorta

In this context, AI becomes the ultimate translator of mathematical thought into action. It is a manifestation of the idea that through mathematics, we can not only comprehend the universe but also manipulate it, creating entities that can operate within the same logical frameworks that govern everything from the orbits of planets to the functioning of ecosystems. AI’s mastery of this language allows it to perform tasks with precision and efficiency that mimic and often exceed human capabilities. Whether it’s analyzing vast datasets to uncover patterns invisible to the human eye or driving vehicles with a level of precision and safety that humans struggle to achieve, AI demonstrates the practical power of mathematics as a language.

  In reflecting on my above phrase about signs and symbols, we see a profound truth about the nature of intelligence — both human and artificial. It is a reminder that at the core of all we endeavor to achieve lies a language that is both ancient and infinitely capable of innovation. Mathematics, with its signs and symbols, is not just a tool for communication; it is the very bedrock upon which the foundation of Artificial Intelligence stands. As we continue to explore and expand the boundaries of what AI can achieve, we do so by delving ever deeper into the mathematical language that underpins the fabric of reality, crafting a future where the line between the created and the creator becomes ever more blurred, united by the universal language of mathematics.

3   From Numbers to Neural Networks

I hope my above discussions have illuminated the indispensable role of mathematics in Artificial Intelligence. As we delve deeper, moving from theoretical foundations to practical applications, it’s crucial to demystify the mathematical notations and symbols that may seem like distant memories from our high school classes. It’s a shift from the philosophical to the practical, a step closer to the very essence of how mathematics intertwines with AI in the real world.

3.1   The Fundamentals

Our exploration begins with the fundamentals — numbers. A concept so basic that even toddlers grasp it in its simplest form. Yet, as we venture deeper, the complexity unfolds. In the realm of computing and programming, numbers transform and transcend their everyday appearances. While our digital companions, the computers, operate within the binary constraints of zeros and ones, the mathematical domain they navigate is infinitely richer. From decimals to negatives, mathematics spans a spectrum far beyond the binary’s simplistic dichotomy. You might wonder, how these machines interpret and manipulate the vast array of mathematical concepts using just two digits. The journey from the binary 101 representation of the decimal number 5 to performing intricate calculations is nothing short of miraculous. It’s a testament to the ingenuity of programming and algorithm design, which translates the rich, complex language of mathematics into a form comprehensible by machines.

  In the realm of Artificial Intelligence, mathematics is not just a tool; it is the very essence that breathes life into the algorithms and models that define AI. As we continue our journey, it’s crucial to understand that AI, in its quest to emulate human intelligence relies on a myriad of mathematical structures and theories. These include but are not limited to, statistics for making predictions, calculus for understanding change, and linear algebra for data representation and manipulation. Consider, for example, the concept of a Neural Network — a cornerstone of modern AI and Deep Learning. At its heart, a Neural Network is a tapestry woven from the threads of mathematical equations, each node and connection representing a mathematical function. The beauty of these networks lies in their ability to learn and adapt, to fine-tune their parameters in response to the data they encounter. This process, known as training, is grounded in calculus and optimization, illustrating how mathematics empowers machines to learn from experience, much like humans do.

3.2   Power of Algorithms and their Sustainability

Furthermore, the role of algorithms in AI cannot be overstated. An algorithm, in its essence, is a set of mathematical instructions or rules that dictate the steps needed to solve a problem. From sorting data to making decisions, algorithms are the brain behind the AI’s ability to perform tasks. The elegance of algorithms lies in their universality. A well-designed algorithm can be applied across different domains, from recognizing speech to diagnosing diseases, showcasing the versatile power of mathematical logic in AI. Yet, the journey from mathematical theory to practical AI application is caught with challenges. The complexity of translating human understanding into a language that machines can interpret involves not just technical prowess but also a deep philosophical understanding of what it means to know or learn. It raises profound questions about the nature of intelligence and the limits of machine learning.

  In addressing these challenges, we also encounter the concept of computational complexity — how we measure the efficiency of algorithms and the resources they require. This area, deeply rooted in mathematical logic, is critical for developing AI systems that are not only intelligent but also efficient and sustainable. As we stand on the tall foundation of a future shaped by AI, it’s clear that our journey through the landscape of mathematics and programming is far from over. It is a path of endless discovery, where each step forward in our understanding of mathematics opens new doors for AI innovation. So let’s start small, right from the basics.

4   Symphony of Numbers in Artificial Intelligence

Before we delve into the complex algorithms and neural networks that form the backbone of AI, it’s essential to grasp the basics of the numbers that are the building blocks of mathematical logic and computation. In the grand orchestra of mathematics, different types of numbers play unique roles, each contributing its distinct sound to the symphony of AI.

4.1   Real Numbers

Real numbers, represented by \(\mathbb{R}\) (called double-struck capital R), play the part of the vast, continuous spectrum that underpins the universe of computation and modeling. Within the realms of Artificial Intelligence and Machine Learning, the significance of real numbers cannot be overstated. They form the very fabric upon which the intricate patterns of data and algorithms are woven, enabling a multitude of applications that span from the simplest of calculations to the most complex of predictive models. I often tell people that real numbers are…

Mathematical entities which capture the continuum of possibility

Real numbers are all the numbers that can be found on the number line, encompassing both the rational numbers (such as \(1/2\) or \(2.\overline{79797979\ldots}\)) and irrational numbers (such as \(\sqrt{2}\) or \(\pi\)). This includes integers and fractions, as well as numbers that cannot be precisely expressed as a fraction of two integers. Real numbers represent quantities along a continuous scale, making them indispensable for measuring and representing quantities in the physical world, such as distance, time, temperature, and probability.

  In AI and ML, real numbers are the cornerstone of data representation. They are used to quantify and encode information about the world, serving as inputs and outputs for models that learn from data. Whether it’s the pixels in an image, the frequencies in a sound recording, or the features of a dataset describing housing prices, real numbers capture the nuances of information in a form that machines can process and learn from. Moreover, real numbers are pivotal in the formulation of models themselves. The weights in neural networks, the coefficients in regression models, and the distances in clustering algorithms are all expressed as real numbers. The optimization algorithms that train these models, seeking to minimize error or maximize accuracy, operate in the realm of real numbers, navigating the complex landscapes of high-dimensional spaces to find the best parameters for the task at hand.

real-numbers.py

π = 3.14159  # An approximation of Pi
e = 2.71828  # An approximation of Euler's number

The role of real numbers in AI and ML is akin to the air that musicians breathe is invisible, yet utterly essential. They allow for the encoding of complex, continuous phenomena in a manner that computational models can understand and act upon. The precision and continuity of real numbers facilitate the modeling of intricate patterns and relationships in data, making possible the development of technologies that can predict, automate, and augment human capabilities. As we continue to explore the various members of the numerical orchestra in AI, the versatility and omnipresence of real numbers underscore their critical role in composing the future of intelligent systems.

4.2   Complex Numbers

Complex numbers, represented by \(\mathbb{C}\) (called double-struck capital C), are an elegant solution to equations that cannot be solved using real numbers alone. A complex number is typically represented as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\). This structure allows complex numbers to express an astonishing range of phenomena, from the oscillations of waves to the behavior of quantum particles. In AI and ML, complex numbers find their use in several advanced algorithms and data processing techniques. For example, they are pivotal in the Fourier transform, a mathematical technique that transforms a function of time (or space) into a function of frequency. This transformation is instrumental in signal processing, enabling computers to understand and manipulate audio, images, and other data types in ways that would be cumbersome, if not impossible, with real numbers alone.

  Moreover, complex numbers facilitate computations in neural networks, particularly those that deal with waveforms or oscillatory data. By encoding information in both the magnitude and phase of a complex number, AI systems can capture patterns and relationships in data that might be missed when using only real numbers. The use of imaginary numbers extends beyond the mere theoretical into the very fabric of algorithms that shape our digital world. In AI and ML, these numbers are not figments of fantasy but tools of immense practical utility. Moreover, in the realm of neural networks — a fundamental building block of machine learning — imaginary numbers contribute to the optimization of complex functions. These networks, akin to a simplified model of the human brain, learn from vast datasets. The incorporation of complex numbers (which include both real and imaginary parts) can significantly enhance the capacity of neural networks to model intricate patterns and dynamics. This is particularly evident in tasks involving sequences and time series data, such as speech recognition, music generation, and predictive modeling, where the temporal dynamics are crucial.

complex-numbers.py

complex_number = 3 + 4j                             # (3+4j)

real = complex_number.real                          # 3.0
imaginary = complex_number.imag                     # 4.0

sum_of_complex_numbers = complex_number + (2 - 3j)  # (5+1j)

Understanding complex numbers and their applications in AI and ML opens up a world of possibilities. From the processing of natural phenomena to the enhancement of algorithms, complex numbers allow us to approach problems from a multidimensional perspective, offering solutions that are as elegant as they are effective. They allow us to venture beyond the limitations of the real number line, embracing a multidimensional perspective that is essential for tackling the complex challenges of today’s technology. Through their use, we can unlock new possibilities and enhance the capabilities of AI systems, making them more powerful, efficient, and versatile.

4.3   Rational Numbers

Rational numbers, represented by \(\mathbb{Q}\) (called double-struck capital Q), are familiar to us from our earliest days of mathematics and play yet critical role in the domains of Artificial Intelligence and Machine Learning. Unlike their mysterious counterparts, the complex numbers, rational numbers might seem mundane at first glance. Yet, their utility and significance in AI and ML are profound, offering a bridge between abstract mathematical concepts and the tangible, quantifiable world we seek to understand and manipulate through technology. Rational numbers are defined as any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(p\) is the numerator, \(q\) is the denominator, and \(q \neq 0\). This definition encapsulates all integers, as every integer \(z\) can be represented as \(\frac{z}{1}\), and extends to fractions that represent precise values such as \(\frac{1}{2}\), \(\frac{3}{4}\), and so on. Rational numbers fill the number line densely, between any two rational numbers lies another, offering an infinity of precision that is both a boon and a challenge in computational contexts.

  In AI and ML, rational numbers serve as the bedrock for representing and processing data. They are crucial in algorithms that require precise, exact calculations, such as those found in operations research, linear programming, and optimization problems where decisions must be made under constraints. Rational numbers help in representing proportions, probabilities, and statistical measures with exactitude, facilitating nuanced analysis and decision-making processes. Furthermore, rational numbers are indispensable in the training of machine learning models, especially in supervised learning where features and labels are quantified. Their precision allows for the accurate measurement of model performance, error rates, and improvements over time, enabling researchers and practitioners to fine-tune algorithms with a high degree of control.

rational-numbers.py

from fractions import Fraction

# Without using Fraction
rational_1 = 1 / 2           # Prints 0.5
rational_2 = 3 / 4           # Prints 0.75

# With Fraction
fraction_1 = Fraction(3, 4)  # Represents 3/4
fraction_2 = Fraction(5, 6)  # Represents 5/6

Rational numbers, with their precise and interpretable nature, act as a foundational element in the vast and varied landscape of Artificial Intelligence. They remind us that amidst the complexity and sophistication of modern AI algorithms, simple and exact mathematical concepts retain a place of importance. Through rational numbers, we can precisely model and solve problems, paving the way for advances that are as grounded in mathematical rigor as they are in innovative thinking.

4.4   Irrational Numbers

Irrational numbers emerge as enigmatic and profound melodies that echo the infinite complexity of the universe. Unlike their more straightforward counterparts — the integers and rational numbers — irrational numbers cannot be expressed as a simple fraction of two integers. Their decimal representations are endless and non-repeating, stretching into infinity without ever settling into a repeating pattern. This category includes the famous constants \(\pi\) (pi), the ratio of a circle’s circumference to its diameter, and \(e\) (Euler’s number), the base of natural logarithms, among others. These numbers play pivotal roles not only in mathematics but also in the realm of Artificial Intelligence and Machine Learning.

  In AI and ML, irrational numbers often represent continuous values in calculations and models. While the precise values of irrational numbers cannot be fully captured, approximations are used to represent concepts such as growth rates (using \(e\)) or geometric calculations (using \(\pi\)). These approximations allow for the modeling of natural phenomena with a high degree of accuracy. Certain algorithms, especially those involving optimization and search techniques, leverage properties of irrational numbers. For instance, the use of \(\pi\) and \(e\) can be found in algorithms that require complex calculations of areas, volumes, or natural growth patterns, enabling more precise and effective models. Irrational numbers also find application in encryption algorithms within the realm of cybersecurity in AI. The unpredictable nature of their decimal expansions can contribute to creating more secure encryption keys, making it harder for unauthorized entities to decipher protected information.

irrational-numbers.py

import math

# Accessing irrational numbers π and e
π = math.pi
e = math.e

Irrational numbers, with their endless, non-repeating decimals, offer a glimpse into the infinite. They contribute depth and complexity, enabling models and algorithms to more closely mirror the intricacies of the world. Their approximations provide the tools needed to explore, model, and understand continuous processes and natural phenomena, making them indispensable in the ongoing quest to advance AI and ML. Through the calculated use of these numbers, we can achieve greater precision and insight, allowing technology to better serve humanity’s quest for knowledge and innovation.

4.5   Integers

Integers, represented by \(\mathbb{Z}\) (called double-struck capital Z), are the quintessence of discrete data representation. In AI and ML, many scenarios necessitate a straightforward, uncomplicated expression of quantity — be it the count of occurrences, indexing of arrays, or the labeling of categories. For instance, in a dataset, each category or class might be assigned an integer value, transforming qualitative data into a form that algorithms can efficiently process and learn from. The bedrock of AI and ML is constituted of algorithms, where integers frequently govern the flow of execution. Whether it’s the number of iterations in training loops, steps in optimization algorithms, or layers in deep neural networks, integers serve as the controlling variables that guide the procedural logic and iterative processes critical to learning from data.

  In the realm of cybersecurity, integral to the safe and ethical use of AI, integers play a key role in encoding and encryption algorithms. The transformation of data into secure formats often relies on integer-based computations, ensuring that information remains confidential and tamper-proof, a paramount concern in AI applications handling sensitive or personal data. When it comes to computational tasks, integers are preferred for their efficiency. Operations on integers are typically faster than those on floating-point numbers, making them advantageous for performance-critical applications in AI and ML, such as real-time processing and analysis where rapid execution is crucial. Python, a lingua franca of AI and ML, treats integers as a fundamental data type, enabling direct and intuitive interaction. Integers in Python are represented simply by numeric literals without a decimal point, accommodating a wide range of values without explicit limits, only bounded by the machine’s memory.

integers.py

# Example of Integers in Python for AI/ML Applications

# Number of features in a dataset
num_features = 64

# Total samples in a dataset
total_samples = 10000

# Samples to be used for training (80% of the total)
train_samples = int(total_samples * 0.8)

# Iterating over a fixed number of epochs for model training
epochs = 10
for epoch in range(1, epochs + 1):
    print(f"Training epoch {epoch}/{epochs}...")

# Calculating the number of samples not used in training
test_samples = total_samples - train_samples
print(f"Training samples: {train_samples}, Test samples: {test_samples}")

Integers are the rhythmic backbone, essential for structuring data, guiding algorithmic processes, and ensuring computational efficiency. Their straightforward, discrete nature makes them indispensable in the orchestration of AI and ML, from the most fundamental tasks to the most advanced computational processes. As we delve deeper into the mathematical melodies that power AI, the role of integers — as the fundamental notes that resonate through the fabric of algorithms — remains ever crucial, shaping the harmony of Artificial Intelligence.

References

1. Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble, https://mathworld.wolfram.com/TranscendentalNumber.html

2. Transcendental number, a number that is not algebraic, in the sense that it is not the solution of an algebraic equation with rational-number coefficients. Transcendental numbers are irrational, but not all irrational numbers are transcendental, https://www.britannica.com/science/transcendental-number

3. In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion, https://en.wikipedia.org/wiki/Real_number

4. The Real Numbers had no name before Imaginary Numbers were thought of. They got called “Real” because they were not Imaginary. That is the actual answer!, https://www.mathsisfun.com/numbers/real-numbers.html

5. In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit, https://en.wikipedia.org/wiki/Complex_number

6. A Complex Number is a combination of a Real Number and an Imaginary Number, https://www.mathsisfun.com/numbers/complex-numbers.html

7. The Fourier Transform is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by the sine and cosine functions of varying frequencies, https://www.thefouriertransform.com/

8. All four members of the Fourier transform family (DFT, DTFT, Fourier Transform & Fourier Series) can be carried out with either real numbers or complex numbers, https://www.analog.com/media/en/technical-documentation/dsp-book/dsp_book_Ch31.pdf

9. We want to transfer the signal from the space or time domain to another domain - the frequency domain. In this domain, the signal has two “properties” - magnitude and phase. If we want to get only the signal’s “power” in a specific frequency bin, we indeed only need to take the absolute value of the Fourier transform, which is real. But, the Fourier transform gives the phase of each frequency as well, https://math.stackexchange.com/questions/275115/why-do-fourier-transforms-use-complex-numbers

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