An easier response is to note that it is your friend that is making the positive claim, so the onus is on him to prove it. There is no obligation on you to provide any evidence to the contrary. You just need to challenge him by saying 'that doesn't sound very plausible to me. What argument do you have to support your claim?'

FWIW, the claim sounds implausible to me.

How do you mathematize it? One way it's been done is this. You ask people a bunch of questions which distinguish between different political beliefs, producing a score on a Likert Scale. The Likert scores can then be given as a representation of your political beliefs, or alternatively you can minimise the distance to a set of ideologies/people who believed them to find who you're politically 'close' to. The political centre in this case is just 'not belonging to any extreme', so is a range of middling scores on the questions.

In other words, he seemed to claim that altough ones political positions might be all over the spectrum in different issues, at the very core one is part of either one ideology or another. — rickyk95

On a non-mathematical, personal basis, I have not found this to be true. When I talk to people across the political spectrum, even those with very different political leanings, I've found that the true break is between people of good will and those who are not. Both can be found at any position along the conservative vs. liberal line except perhaps at the extremes.

You mathematize it sloppily.

First you establish a spectrum with two perceivably distinct poles (i.e authoritarianism v libertarianism, socialism v capitalism, democracy vs technocracy, conservatism vs progressivism, etc...)

Then you ask people a bunch of questions about a range of political topics which gauge which side they're on concerning the specific spectrums you've identified.

Then you just count how many times they took a particular stance on issues vs how many times they took the opposite stance, and the ratio (or difference, depending on the approach) will represent their displacement from center on the political spectrum.

It's sloppy work rife with inaccuracy, but then, so too are our individual political opinions...

It's not as if everyone's political views neatly and consistently flow from a set of core principles, which is why your friend is somewhat wrong Sometimes we just make shit up on the spot.