If you're a player in a roleplaying game, you're pretty happy when you get to add a number to your character's attribute or skill. Your character becomes [potentially] more powerful, which is nice because the fantasy of being infallibly good at something is one of the fun things about most roleplaying games. In fact, when numbers on a character sheet stagnate, it's usually seen as a sign that progress has stopped, which in turn means it's probably time for a fresh character. The game is over. This poses an interesting design challenge, because while numbers can literally keep getting bigger forever, it's also impractical. Dnd 5e and Pathfinder 2 each meet this design challenge in some interesting ways, and I think there's a lot to learn from the different approaches.
Whether it's Tales of the Valiant or Dungeons & Dragons (2014) or some other 5e variant, the 5e approach to numbers is highly constrained. I think when people say things like "the math in 5e is tight" or "5e is elegant" and so on, they're actually referring to is the fact there's a fixed scale for success and failure. The fixed scale is seen as being 1 to 20 (it's actually more like 1 to 30 or 35), which ensures that the rest of the game's design only confers little bonuses to character stats, so players are rarely adding more than 10 to their roll.
Additionally, the game master only has to learn to judge success or failure once, because roll results rarely go above 20. A result of 5+ is good enough for an easy task, 10+ is good for a challenging tasks or to hit a monster, and 15+ achieves the improbable or hits a big actually-scary monster.
In 5e, this is all made possible by at least 3 significant design choices.
In 5e, you only get +1 to your Proficiency Bonus every 5 levels. When that happens, it affects all skills you have Proficiency in. That feels like a pretty major upgrade when it happens, but it only happens every 5 levels.
Parsing out this bonus so it happens only every 5 levels is not insignificant. For some players, it's delayed satisfaction, and a way to reinforce the pursuit of those delicious experience points (XP). For other players, it's makes leveling up not very meaningful, and a little less fun than it could be. It depends on the player, but in the end it does contribute to that core mechanic of gaining XP to level up.
If you don't have proficiency in a skill, there's no way to make its numbers better aside from improving its parent attribute score. That in itself can be frustrating, but there are subsystems to help players gain proficiency in skills they didn't get during character creation.
Only 1 attribute score gets a meaningful bonus, every other level. (Actually you can split the bonus between 2 attributes, but then the modifier is less likely to go up, so it's safe to say that 1 attribute improves every other level.) An increase of 2 to an attribute score earns a +1 modifier to its roll, and that modifier affects all of its derivative skills, so that also feels really good when it happens.
As with proficiency boosts, the slow pace can frustrate some and delight others. Unlike proficiency, there's an optional added decision that allows a player to choose a feat rather than an attribute increase, and that can help disguise the slow pace. The more you have to ponder a decision, the more satisfying it is to have made a decision.
Part of creating a fixed scale is to ensure that the scale is only needed for a fixed amount of time. Dnd has developed into a system that supports 20 levels, regardless of a character's lineage or class, so the fixed scale of failure to success from 1 to 20 has to last for 20 levels. After that, the numbers do stop going up on character sheets, implying that the game is over. Then it's time to retire your character, and build a new one at level 1.
At level 20, a character could well have a +5 to at least one attribute score and a +5 proficiency bonus. That's a clean +10 bonus to many rolls, which means a roll of 30 is possible on a d20. With a +10 to a roll, it's basically impossible to fail (unless you roll a 1) at any task that's been established as normal or common. An action with a DC 15 is trivial, and even DC 20 doesn't seem very threatening. It's rare to see a DC 30 or above in 5e, so at high levels the player characters have truly tipped the fixed scale in their favour. Accordingly, it's pretty common for game masters to complain about running high level 5e games.
There have been hints that, but for tradition, 5e could have been released as a system supporting levels 1 to 10 only. Mike Mearls, co-lead designer of 5e, has said that the design team noticed that most adventuring parties stop playing at level 14 or so. The 5e system itself appears to be conflicted. The system supports up to level 20, and yet few (if any?) official 5e adventures have taken characters all the way to level 20.
Mike has also said that adventuring parties should level up faster. I feel that the official 5e adventures mostly reflect this. Modules like Princes of the Apocalypse and Out of the Abyss and Curse of Strahd progress from level 3 or 8 to 14 or so, and often recommend an increase in level at the end of each chapter.
The implementation, if not the design, of 5e appears to encourage players to level up often to satisfy the hunger for stat increases, and to get to the end of the adventure so you can create new characters and start the process over again. I'm not being cynical, but pragmatic, when I assume that Wizards of the Coast probably had the intention of persuading players to spend less time meandering through each adventure and to spend more focus time getting the adventure done so they could buy the next book. Personally, I see their point. To date, the most satisfying 5e adventure for me as a player and then as a games master has been Out of the Abyss, which I played and ran as a speed run in one week of playing every evening. I run Expedition to Castle Ravenloft every Halloween for a single 8-hour session. There are too many fun adventures out there to let every book you buy become a 5 year campaign, and too little time to spend idly chatting in an imaginary fantasy land when you could go out and find treasure and slay monsters instead.
When a player gains a new item that grants a bonus, it feels like a stat increase even though the increase is not innate to the character. Bonuses from equipment are minimal, and there are rules to ensure a character can't use a bunch of different items with lots of bonuses all at once. In practise, equipment adds a potential +5 in bonuses to the design space, which is easy to account for.
I noted before that at level 20, a character could have a +5 attribute modifier plus a +5 proficiency bonus for a base +10 bonus to many rolls. Add in a few magic items and you could have +15 to certain rolls. That kind of bonus does strain the fixed scale of 1 to 20, and that's an acknowledged weakness of the system, but the game's design does much to mitigate the scope.
First of all, magic attunement rules ensures that no player has more than 3 magic items active at any given time. Also, many magic items don't confer direct stat bonuses and instead are circumstantial. And finally, when magic items do confer a modifier, it's usually to a specific skill or even a specific action.
Players love to see stats increase, and Pathfinder 2 grants their every wish by embracing big numbers. For example, you don't reach +4 proficiency in 5e until level 5, but in Pathfinder 2 you can get to +4 (the Expert rank of proficiency) by level 1, depending on what kind of character you're building. That's not to say that every 1 level in Pathfinder is equal to 5 levels of 5e, but the Pathfinder scale is definitely not calibrated on the same 1 to 20 scale as 5e.
In Pathfinder, you can have a dungeon that quite simply out-levels player characters. A location can have challenges with DCs that exceed what is possible for a player character until a certain level. That option basically doesn't exist in 5e, because no matter what the 5e scale starts at 1 and peaks at 30 or 35. The Pathfinder system gives the game master the ability to design challenges meant for a specific player level.
The upper limit of success in Pathfinder is not a fixed point, the way it is in 5e. Instead, Pathfinder allows for critical success on any result that's 10 above its target, and a critical failure on results 10 below target.
That means there's forever a range of 10 numbers that could either make or break your hero's attempt at being heroic. Finally feeling powerful enough to physically pry open that magical lock that nobody's been able to open throughout the entire campaign? Well you might be capable of rolling a 50 on your d20, but what if you roll a 30 instead?
In Pathfinder, Proficiency is not a global value. Just because you increased your proficiency in assessing the value of antique furniture does not mean you've also increased your proficiency, in equal amounts no less, in swordplay. Nearly everything in Pathfinder 2 has a proficiency ranking, from Trained to Expert to Master to Legendary.
Leveling up feels like leveling up, not just because you're making numbers bigger on your character sheet, but because you're having to choose which numbers to increase. You've got a budget of points you can spend, and the longer your character is active the more cool stuff you have to consider. Important decisions like this make the leveling-up process feel substantial.
As with 5e, Pathfinder is able to leverage items for spare modifiers. Items are nice because they feel powerful while you have them, you lose sleep over getting them taken away from you, and if you keep looking you might even find a better version later on.
Powerful items are such an elegant solution to player character boosts. Not only is the power dependent on the item (which can have its own limitations built-in), but also items of great power are a perfect fit for fantasy lore. There's no greater fantasy archetype than an artefact or relic that holds mysterious power within it, and there's no greater trap than to gift one to a player character only to reveal later that the item's also been cursed all along.
Dnd 5e and Pathfinder 2 share a common lineage, and now that they've diverged sufficiently they've both developed unique strengths. Comparing the ways solve common problems can be educational, and lately I think I've been learning the most from the way they use math. It's a technical detail that has little to no actual effect on what the games can do, or even (for the most part) how you experience the games, and yet it makes them very different roleplaying games from a design perspective.