When things become undecidably complex Within linear, one- dimensional thinking. 1. A is A Law of Identity Law of Identity 2. A must be either A or not A Law of Contradiction Law of Contradiction 3. A cannot be both A and not A Law of Excluded Middle Law of Excluded Middle But what if we have n- dimensional thinking along n-lines? But what if we have n- dimensional thinking along n-lines? 0-Dimension 1-Dimension 2-Dimensions 3-Dimensions 4-dimensions From I (a point: 0-D {nothingness, emptiness}) to Other (a line: 1-D {linearity, bivalence}) to I Other (a plane: 2-D {possibly 3-valued}) to Community (a cube: 3-D {many-valued}) to Cosmos (a hypercube: 4:D {potentially -valued}) Of, if you wish, a 4-D Tessaract Now In other words, there is no absolute priority, and everything is possible, given fluctuating times and places and conditions. In other words, there is no absolute priority, and everything is possible, given fluctuating times and places and conditions. Or, consider a sort of Gdels proof, from a concrete, practical point of view Gdel first thought that his theorems established the superiority of mind over machine. Gdel first thought that his theorems established the superiority of mind over machine. Later, he came to a less decisive, conditional view: if machine can equal mind, the fact that it does cannot be proved. Later, he came to a less decisive, conditional view: if machine can equal mind, the fact that it does cannot be proved. This view parallels the logical form of Gdels second theorem: if a formal system of a certain kind is consistent, the fact that it is cannot be proved within the system. This view parallels the logical form of Gdels second theorem: if a formal system of a certain kind is consistent, the fact that it is cannot be proved within the system. Gdels more famous first theorem says that if a formal system (or a certain kind) is consistent, a specific sentence of the system cannot be proved within it. Gdels more famous first theorem says that if a formal system (or a certain kind) is consistent, a specific sentence of the system cannot be proved within it. Extrapolating from the above, in everyday situations, if our knowing becomes inconsistent (flawed, unacceptable), then we go on to choose between all the possibilities that may present themselves, and we proceed to articulate the consequences of our choice at some particular time and place within some particular context. Extrapolating from the above, in everyday situations, if our knowing becomes inconsistent (flawed, unacceptable), then we go on to choose between all the possibilities that may present themselves, and we proceed to articulate the consequences of our choice at some particular time and place within some particular context. But then we cant know if this choice will prove consistent at/in any and all times, places, and contexts, for if it is indeed consistent, nevertheless, there is no knowing whether at some other time, place, and context it will not reveal an inconsistency. But then we cant know if this choice will prove consistent at/in any and all times, places, and contexts, for if it is indeed consistent, nevertheless, there is no knowing whether at some other time, place, and context it will not reveal an inconsistency. Thus whatever choices we make from the range of all possibilities will be incomplete, for there is no way to know that they will not be subject to some inconsistency or another at some particular time and place and within some particular context. Thus whatever choices we make from the range of all possibilities will be incomplete, for there is no way to know that they will not be subject to some inconsistency or another at some particular time and place and within some particular context. Stated otherwise Stated otherwise What good is rigorous formalization that can prove a sentence which says that it is not provable (first theorem)? What good is rigorous formalization that can prove a sentence which says that it is not provable (first theorem)?And, What good is such formalization that can prove its consistency when it would follow that it is not consistent (second theorem)? What good is such formalization that can prove its consistency when it would follow that it is not consistent (second theorem)? Or, in everyday life situations Or, in everyday life situations What good is knowing, if it can know a sentence within itself revealing that it is not knowable (much like Gdels first theorem)? What good is knowing, if it can know a sentence within itself revealing that it is not knowable (much like Gdels first theorem)? What good is knowing that claims to know its own consistency when it follows that that knowing is inconsistent (much like Gdels second theorem)? What good is knowing that claims to know its own consistency when it follows that that knowing is inconsistent (much like Gdels second theorem)? Knowing is either inconsistent or incomplete, or in the worst of all possible worlds, both. Knowing is either inconsistent or incomplete, or in the worst of all possible worlds, both. That is to say: We never know entirely what we mean by what we say or write, or the implications of our interpretation regarding what somebody else says or writes. Nonetheless, we will undoubtedly continue on, in learned ignorance, blissfully saying and writing and listening and reading. That is to say: We never know entirely what we mean by what we say or write, or the implications of our interpretation regarding what somebody else says or writes. Nonetheless, we will undoubtedly continue on, in learned ignorance, blissfully saying and writing and listening and reading. For, concrete everyday living knows of no carved in stone linearity: it thrives on far-from- equilibrium conditions, which allow it always to be in the process of becoming something other than what it was becoming Indeed, I predict a time when threre will be mathematican investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency (Ludwig Wittgenstein, Philosophical Remarks, Blackwell, 1964, p. 332).