Tat Tvam Asi, the Sanskrit equation that we may translate to “That Thou Art”, may be the greatest mathematical statement of the ancient philosophical texts of India, the Upanishads. Its power and profoundness are recognized in its subtle treatment of the concept of a boundary.
Tat indicates ultimate reality—an intimate reality that has no boundaries whatsoever.
Tvam initially indicates you and I, individuals defined by many boundaries, whether it be the boundary of a physical body, the boundary of our thoughts and personalities, or the boundary of identity itself.
Asi indicates absolute equality.
We may write the same statement mathematically as Tat = Tvam. This necessarily appears as a contradiction to the individual mind, which asks,
How can boundlessness equal boundedness when boundlessness is itself the absence of any boundary?
The individual mind’s tendency is to re-write the equation as Tat > Tvam, since no matter how large a boundary we create around Tvam, Tat will always remain prior to and independent of Tvam. This represents the mindset of a seeker, which sees Reality or Truth as something different and bigger than her/himself.
In equating Tat and Tvam, the Upanishads are declaring the apparent difference between boundedness and boundlessness to be superficial. In other words, the equation Tat = Tvam negates the independent reality of boundaries. As the individual mind meditates on Tat and Tvam, the mind expands and becomes subtler. It begins to recognize its complete dependence on Tat, which is prior to the experience of a bounded mind. The subtler the mind becomes, the more the truth of the equation is recognized.
When the bounded mind leaps from its previous understanding of Tvam as the bound individuality to a new recognition of Tvam as Tat itself, the great equation has served its purpose, and is replaced by the final mathematical equation Aham Brahmasmi, I = Brahman. In this equation, I no longer refers to boundedness but rather to I-as-boundlessness, and Brahman refers to ultimate reality.
Thus, Tat Tvam Asi represents a meditative process that results in a transformation of Tvam.
In this process, the negation of boundaries is not blind. The very fact that that the words on this screen have distinct meaning is evidence that we can experience boundaries in our world. In fact, the world itself is the experience of a series of boundaries. Tat = Tvam describes the emergence of this possibility with precision, revealing that the experience of any boundary is directly proportional to the experience of the individual Tvam. In other words, the more the individual-boundary is experienced, the more the mind is in a bounded state, and the more it reflects a world of boundaries and distinct things. Tinted sunglasses invariably reveal a tinted world.
This offers insights into mathematics because mathematics, like all systems of thought, begins with creating boundaries, since the individual mind reflects its bounded nature in the form of thought. The number 1, for example, has explicit boundaries that differentiate it from 0 and 2. This is precisely why the ancient philosophy of India is referred to as Advaita, or non-duality. It is not oneness because oneness is a philosophy that embraces the boundary of all-in-one. Neither is it multiplicity because multiplicity embraces the boundaries of many. Rather, Advaita negates the independent reality of all boundaries, whether one or many, while recognizing that boundaries of many kinds can be superimposed and experienced, yielding many subsets of philosophies.
In the absence of recognizing the intimately personal and comprehensively universal significance of boundaries, it is not possible to understand the nature of the world beyond a certain level. This is why there is much argument today over the implications of quantum physics, which seems to describe the boundaries of fundamental things like particles as changing and fluid. Such findings are registering the state of a fluctuating mind, not an external, independent world.
Thus, understanding the meaning of the mathematical equation Tat Tvam Asi not only delivers liberation from boundedness, it delivers insight into the mechanics of mathematics, physics, and the world itself.