Mathematical thinking vs Physics based thinking - early adopter version of my book
The image is generated by OpenAI/ChatGPT. It shows Einstein because recently, Einstein's theory of plunging regions around a black hole was proven physically. It was hitherto only a mathematical concept - an idea that underpins this article.
This longish article is a part of my forthcoming book on Mathematical foundations of data science. If you want to be a part of the early adopter trial please see this early adopter version of my forthcoming book - mathematical foundations of data science.??
Background
Physics based thinking ??is valued by engineers and engineering in general. Its also more easily understood because it is concrete. In contrast, mathematical thinking is abstract.
In this article, I hope to explain mathematical thinking from the perspective of physics based thinking.
I think the perspective of mathematical thinking is sadly missing in education today. Much of what I am writing here is self taught - mainly to teach my students #universityofoxford. If you want to study AI with me, please see my university of Oxford course on AI
What do we mean by an abstract concept and the process of? abstraction
When we say mathematical thinking is more abstract - what exactly do we mean?
An abstract concept
An abstract concept is an idea or notion that does not have a physical or tangible form. It represents something that exists on a mental, theoretical, or symbolic level rather than in the physical world. Abstract concepts can be difficult to directly observe or measure because they rely on mental understanding rather than physical experience.
?Mathematical concepts like infinity, probability, or imaginary numbers are abstract concepts.
In general, abstract concepts are those that you understand through thought, language, or reasoning rather than through the five senses. They are often the building blocks of theories and frameworks and are essential for discussing complex or universal ideas.
The process of abstraction
There is a related concept called the process of abstraction.??
The process of abstraction is a mental technique used to simplify complex systems by focusing on the most essential features or characteristics of an idea, concept, or object while ignoring the less relevant details.??
Abstraction involves? identifying core elements of a phenomenon, creating generalizations from specifics,? creating a taxonomy or layering information.?
The process of abstraction makes it possible to construct theories, models, and systems that capture the essence of complex ideas, making it easier to work with them without being bogged down by unnecessary details.
Darwin’s theory of evolution as a process of abstraction?
You can think of abstraction as creating a supercategory or creating a model of a phenomenon.? We can think of Darwin’s theory of evolution as a process of abstraction.?
Darwin observed various biological phenomena—variations in finches’ beak shapes, for instance, or the gradual adaptations seen across different environments. He abstracted a general principle from these specific instances, that species evolve over time through natural selection, where traits advantageous for survival are passed down and proliferate. He then evolved the theory into a framework - which applied not just to finches in the Galapagos islands ?but all species across time - all based on specific observations of finches in a specific place(the Galapagos islands)
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How is mathematics related to the process of abstraction
Mathematics is deeply related to? the process of abstraction.? Many ideas in maths use abstraction.?
Mathematics abstracts concepts from specific, real-world examples to create general ideas that apply across situations. For example, the concept of a "number" abstracts from counting objects to representing quantities more broadly, allowing us to create categories like whole numbers, fractions, and imaginary numbers.
Abstraction in mathematics also leads to the creation of structures (like sets, groups, or spaces) and relationships (such as functions or operations) that hold independently of the objects they describe.??
?Mathematics uses abstraction to create models of real-world phenomena, stripping away non-essential details to focus on core relationships and behaviors. For instance, calculus abstracts the concept of change, allowing us to model motion, growth, and decay without specifying each real-world detail.
Finally, you can formulate high level theories? using mathematics.??
Difference Between Mathematical Thinking and Physics-Based Thinking
With this background, we can now contrast mathematical thinking vs Physics based thinking - with a view to understanding mathematical thinking better.
Mathematical thinking and physics-based thinking share similarities.? However, there are important differences
Mathematical Thinking: Focuses on abstract structures, patterns, and relationships. The problems in mathematics are often self-contained, meaning they arise from within mathematics itself, such as proving theorems, solving equations, or analyzing logical structures. The solutions to these problems don't necessarily need to be connected to the physical world; they are often theoretical or abstract.? For example, proving that there are infinitely many prime numbers is a purely mathematical problem, concerned with the internal properties of numbers. In contrast, physics based thinking centers on understanding the real world and how it behaves. Physics problems are grounded in the physical universe, and physics-based thinking involves applying mathematical models and logic to describe, predict, or explain natural phenomena. For example, understanding? how planets move in their orbits involves using Newton's laws of motion, which are based on empirical observations of the physical world.
Mathematical Thinking deals primarily with abstract entities that do not have direct physical representation (like numbers, sets, functions, and geometrical shapes). It’s about discovering truths within these abstract systems, independent of whether they have real-world applications. A mathematician might study higher-dimensional spaces, even though humans can’t directly observe more than three spatial dimensions.
Mathematical Thinking: seeks certainty.? In contrast, Physics-Based Thinking works with approximations. In mathematics, a proof guarantees a result with absolute certainty. Once a statement is proven, it is permanently true within its mathematical framework. In number theory, if a property of numbers is proven, it applies universally and will never be disproven in its domain of validity. In contrast, Physics often deals with complex, real-world systems where exact answers are not always possible, so physicists rely on approximations and models that are "good enough" to explain or predict phenomena, while recognizing that these models may not capture every detail. ?For example, the equations used to predict the trajectory of a spacecraft are approximations that depend on various factors like gravitational pull, which might vary slightly depending on conditions.
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Mathematical Thinking constructs abstract models that may or may not have physical counterparts. These models are based on internal consistency and logical coherence. The beauty and value of a mathematical model come from its elegance and internal logic, rather than its applicability to the physical world.
?A mathematician might explore the concept of a four-dimensional shape (a tesseract or a hypercube), which doesn’t exist in the physical world but is fascinating from a purely mathematical perspective. Physics-Based Thinking: constructs models to represent real-world systems. These models are typically grounded in observation and aim to predict or explain how things work in nature. The focus is on how well the model describes actual phenomena, even if it involves simplifications. The model of an atom as a nucleus surrounded by orbiting electrons is a simplification, but it’s useful for explaining chemical reactions and other phenomena.
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?Data Science has elements of both maths and physics based thinking
The interesting part is: data science has elements of both mathematical (ex modelling) and physics based systems (ex approximation).??
?This? article is a part of my forthcoming book on Mathematical foundations of data science. If you want to be a part of the early adopter trial please see this early adopter version of my forthcoming book - mathematical foundations of data science.??
?If you want to study AI with me, please see my university of Oxford course on AI
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Enabling Digital Transformation across the enterprise via process and technology disruptions.
4 个月Physics based thinking is what has been referred to as Applied Maths or as per the UK A Level curriculum, Mechanics. Modelling of data from a Data Science perspective is still Applied Mathematics as the data originates from the real world. The use of data structures such as sets, vectors, tuples, etc involving real and complex numbers is the process of transformation of the raw data into a format suitable for algorithmic analysis. A classic example is that of the Fourier Transform without which signal processing would be impossible. Further conversion of the spectrum of small overlapping time windows of an audio signal into the cepstrum allows for modelling of phonetic components of a word that can now be combined into temporal probabilistic state machines using techniques such as Hidden Markov Models. Each concept described above is mathematical in its origins but is best suited to solve real world problems. Just like theoretical physics has allowed for making predictions that maybe eventually proved or disproved by engineering driven experimentation in the future, the underlying mathematics was initially considered to be abstract until someone came along to figure out how to apply it to a real world problems.
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4 个月Great advice
Generative AI Pioneer I CEO GoMicro
4 个月Interestingly, this differensiation is ingored in high school education, where a soup of formulas is fed to students without differentiating mathematical concepts from physics concepts. I wonder if it is possible to teach abstract math concepts purely based on physical observations, as this is something that students can see. For example, if teach the sine graph from the swing of a pendulum rather than a sine graph first? I wonder if a physics-first approach would be better for building mathematical abstraction?
Digital Twin maker: Causality & Data Science --> TwinARC - the "INSIGHT Digital Twin"!
4 个月????
Lead/Staff Software Engineer | Spirited Entrepreneur | Innovator
4 个月Very well put! This definitely answers one of the questions I posed the other day to you! Thank you, for listening and sitting through each of our conversations Sir! Absolutely thrilled and honored as I keep saying :) learning a lot from your book as well! :)