Grassmann: Space Defined
Hermann Grassmann was born in 1809 in Germany and died in 1877. He was not part of the famous mathematical circles of his time. He was a school teacher, working quietly, far from the major centers of research. Yet in his isolation, he began
Hermann Grassmann was born in 1809 in Germany and died in 1877. He was not part of the famous mathematical circles of his time. He was a school teacher, working quietly, far from the major centers of research. Yet in his isolation, he began thinking about something fundamental that no one had fully understood before. What exactly is space? Until then, space was something physical. It was what we see around us. Length, width, height. Three dimensions. Geometry was tied to intuition and physical experience. Points, lines, and shapes existed in the world, and mathematics described them. Grassmann took a completely different approach. Imagine this. You are standing in a room and you take a step forward. That movement has a direction and a magnitude. Now take another step, but this time to the side. That is another direction. Now imagine combining these two movements. Instead of thinking about where you are, think about the movements themselves. Each movement becomes an object. Now go further. Imagine you are no longer restricted to just three directions. What if you could define a space with four directions, or five, or infinitely many? You cannot visualize it, but you can still describe it. Each direction behaves like a building block, and combinations of these directions create a structure. Grassmann realized something profound. Space does not need to be tied to physical intuition. It can be defined abstractly, using independent directions that can be combined in consistent ways. These directions are what we now call vectors, and the structure they form is what we now call a vector space. This idea is revolutionary because it separates space from physical reality. Space becomes something that can exist in mathematics even if it cannot be visualized. It becomes a framework where relationships matter more than appearance. To understand how deep this is, imagine describing motion, forces, or even data using these abstract directions. You no longer need a physical picture. You only need consistency in how these directions combine. This allows mathematics to move beyond the visible world into higher dimensions that can still be reasoned about precisely. Grassmann also introduced the idea that some directions are independent, meaning they cannot be created from combinations of others. This becomes the foundation of linear independence, one of the most important concepts in modern mathematics and physics. It tells us how to build a space from its simplest components. His work was far ahead of his time. When he published his ideas, they were not widely understood. Many mathematicians could not see the value in a theory that described spaces that could not be visualized. His work was largely ignored during his lifetime. But later, mathematicians realized what he had done. He had created a language for describing space in its most general form. A language that does not depend on physical intuition, but on structure and consistency. Today, this idea is everywhere. Physics uses it to describe forces and fields. Computer science uses it to represent data in high-dimensional spaces. Machine learning models operate in spaces with hundreds or thousands of dimensions. None of this would be possible without the idea that space can be abstract. What makes Grassmann’s thinking extraordinary is that he detached space from reality in order to understand it more deeply. He showed that what we perceive as space is just one instance of a much larger concept. He did not just describe space. He redefined what space could be.