Symmetry arguments
Symmetry arguments are a type of argument that shows up in several fields. There are examples from maths to physics to philosophy, which all share a basic format.
What is symmetry?
The definition I will be using for the word “symmetry” is a way to transform an object that preserves certain properties of the object that we care about. This generalises the everyday definition of symmetry: e.g., a symmetric face is one that can be reflected in the vertical axis (the transformation) and retain its appearance (the property we care about). A shape with a rotational symmetry is one that be rotated and still appear the same.
Probability
I flip a coin and pull three balls out of a hat, independently. The probability of getting heads and three red balls is 0.43. Then, what is the probability of getting tails and three red balls (assuming that the coin is fair)?
Well, the decision of which side of the coin to label as heads or tails was arbitrary because they occur with equal probability and everything else is independent. Therefore, we could take any true sentence about the setup here, then exchange every occurrence of “heads” with “tails” (and vice versa) and end up with another true sentence. This is a symmetry: the transformation is replacing the word “heads” with “tails” and the property we preserve is truth.
So, in the statement above, interchange the word "heads" with "tails", to get that the probability of getting tails and three red balls is also 0.43.
Philosophy
Pascal’s wager is an argument due to (you guessed it…) Pascal. It argues in favour of acting as though his Catholic conception of God is true. The argument goes as follows. If God exists, the payoff of acting according to his will is large. The payoff of disobeying his will would be extremely negative. But if God does not exist, we lose little either way. Therefore, given uncertainty as to whether God actually does exist, we ought to play it safe by acting as though he does. In Pascal’s words:
Let us weigh the gain and the loss in wagering that God is… If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.
On the surface, this is compelling. If we are to be rational, could we gamble against God, risking an eternity of hell and forgoing the ultimate reward of heaven?
But symmetry allows us to diffuse this argument. For every god we can posit an anti-god, where the anti-god sends us to heaven precisely if the god would have sent us to hell. Then, the payoffs of acting as though the anti-god exists are precisely the inverse of the payoffs of acting the same way given that the god exists! This is a symmetry in the space of Gods. Since Pascal’s wager then compells us to act both as if the God and the anti-God exist, this leads us to a contradiction and we can reject it.
A theist could respond to this argument by arguing that in fact Pascal’s wager does work because the probability of their god existing is higher than the probability of the corresponding anti-god, as evidenced by the fact that their god has followers and an established religion behind it. This could be countered using the “many gods” response, that in practice there are many religions with different gods that have opposing wills. This then weakens the wager for any given god.
Physics
Physical theories are often constrained to be symmetric various ways, such as translations or rotations in space, translation or reversal of time, and exchanging particles with their antiparticle. This is a very intuitive property for a physical theory to have: if we move everything in our system forwards by 5 metres, we wouldn’t expect its evolution forwards in time to change (everything would play out the same, just 5 metres away). Equally, we don’t expect the laws of physics to be different if the events occur five years into the future. We should be able to make modifications to the inputs of our physical laws and have the dynamics be invariant: this is the symmetry.
This then allows some problems to be simplified.
Group theory
Group theory is the branch of mathematics that formalises this idea of symmetry. Inuitively, a collection of symmetries form a group if they satisfy the following properties:
One of the symmetries is the symmetry which just leaves everything alone (the identity)
For every symmetry, there is a another symmetry in the group which results the identity if their transformations are performed consecutively (the inverse symmetry)
The group is closed: any time you take two symmetries in the group, you can obtain a new symmetry in the group by performing their transformations consecutively (potentially depending on the order)
Group theory then looks at groups in the abstract. A celebrated result is the classification of finite simple groups, which has a proof that spans tens of thousands of pages and five decades. Noether’s theorem formalises a connection between symmetry groups and conservation laws in physics.